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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
@[simp]
lemma dual_id : @dual d 1 = 1 := by
simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
@[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
simp only [dual, transpose_mul]
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
· noncomm_ring [minkowskiMatrix.sq]
· noncomm_ring
@[simp]
lemma dual_dual : dual (dual Λ) = Λ := by
simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
trans (η * η) * Λ * (η * η)
· noncomm_ring
· noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_eta : @dual d η = η := by
simp only [dual, eq_transpose]
noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
noncomm_ring
@[simp]
lemma det_dual : (dual Λ).det = Λ.det := by
simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
group
norm_cast
simp
@[simp]
lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,
mulVec_mulVec, ← mul_assoc, minkowskiMatrix.sq, one_mul, (vecMul_transpose Λ x).symm, ←
dotProduct_mulVec]
@[simp]
lemma dual_mulVec_left : ⟪(dual Λ) *ᵥ x, y⟫ₘ = ⟪x, Λ *ᵥ y⟫ₘ := by
rw [symm, dual_mulVec_right, symm]
lemma matrix_apply_eq_iff_sub : ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ ↔ ⟪v, (Λ - Λ') *ᵥ w⟫ₘ = 0 := by
rw [← sub_eq_zero, ← LinearMap.map_sub, sub_mulVec]
lemma matrix_eq_iff_eq_forall' : (∀ v, Λ *ᵥ v= Λ' *ᵥ v) ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ := by
refine Iff.intro (fun h ↦ fun w v ↦ ?_) (fun h ↦ fun v ↦ ?_)
· rw [h w]
· simp only [matrix_apply_eq_iff_sub] at h
refine sub_eq_zero.1 ?_
have h1 := h v
rw [nondegenerate] at h1
simp only [sub_mulVec] at h1
exact h1
lemma matrix_eq_iff_eq_forall : Λ = Λ' ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ := by
rw [← matrix_eq_iff_eq_forall']
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact fun _ => congrFun (congrArg mulVec h) _
· rw [← (LinearMap.toMatrix stdBasis stdBasis).toEquiv.symm.apply_eq_iff_eq]
ext x
simp only [LinearEquiv.coe_toEquiv_symm, LinearMap.toMatrix_symm, EquivLike.coe_coe,
toLin_apply, h, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton]
| lemma matrix_eq_id_iff : Λ = 1 ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, w⟫ₘ | minkowskiMetric.matrix_eq_id_iff | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
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"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
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"Mathlib.Order.CompactlyGenerated.Basic",
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"Mathlib.Data.DFinsupp.Basic",
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"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.LinearAlgebra.Pi",
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"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
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"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
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"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
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"Mathlib.LinearAlgebra.FinsuppVectorSpace",
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"Mathlib.RingTheory.Congruence",
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"Mathlib.RingTheory.Finiteness",
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"Mathlib.RingTheory.Nilpotent",
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"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
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"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
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"Mathlib.Data.Fintype.Perm",
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} | {
"hasProof": true,
"proof": ":= by\n rw [matrix_eq_iff_eq_forall]\n simp",
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} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`.
We define the give a series of lemmas related to the determinant of the Lorentz group.
-/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
open minkowskiMetric
variable {d : ℕ}
/-- The determinant of a member of the Lorentz group is `1` or `-1`. -/
lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
refine mul_self_eq_one_iff.mp ?_
simpa only [det_mul, det_dual, det_one] using congrArg det ((mem_iff_self_mul_dual).mp Λ.2)
local notation "ℤ₂" => Multiplicative (ZMod 2)
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
instance : DiscreteTopology ℤ₂ := by
exact forall_open_iff_discrete.mp fun _ => trivial
instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
instance _root_.Finite.instDiscreteTopology [TopologicalSpace X] [T1Space X] [Finite X] : DiscreteTopology X :=
discreteTopology_iff_forall_isClosed.mpr (· |>.toFinite.isClosed)
/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
@[simps!]
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
toFun x := if x = ⟨1, Set.mem_insert_of_mem (-1) rfl⟩
then Additive.toMul 0 else Additive.toMul (1 : ZMod 2)
continuous_toFun := continuous_of_discreteTopology
/-- The continuous map taking a Lorentz matrix to its determinant. -/
def detContinuous : C(𝓛 d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨Λ.1.det, Or.symm (LorentzGroup.det_eq_one_or_neg_one _)⟩,
continuous_toFun := by
refine Continuous.subtype_mk ?_ _
exact Continuous.matrix_det $
Continuous.comp' (continuous_iff_le_induced.mpr fun U a => a) continuous_id'
}
@[simp]
lemma _root_.toMul_eq_one {α : Type*} [One α] {x : Additive α} :
Additive.toMul x = 1 ↔ x = 0 :=
Iff.rfl
lemma detContinuous_eq_one (Λ : LorentzGroup d) :
detContinuous Λ = Additive.toMul 0 ↔ Λ.1.det = 1 := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
Subtype.mk.injEq, ite_eq_left_iff, toMul_eq_one]
simp only [toMul_zero, ite_eq_left_iff, toMul_eq_one]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· by_contra hn
have h' := h hn
change (1 : Fin 2) = (0 : Fin 2) at h'
simp only [Fin.isValue, one_ne_zero] at h'
· intro h'
exact False.elim (h' h)
lemma detContinuous_eq_zero (Λ : LorentzGroup d) :
detContinuous Λ = Additive.toMul (1 : ZMod 2) ↔ Λ.1.det = - 1 := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
Subtype.mk.injEq, Nat.reduceAdd]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· by_contra hn
rw [if_pos] at h
· change (0 : Fin 2) = (1 : Fin 2) at h
simp only [Fin.isValue, zero_ne_one] at h
· cases' det_eq_one_or_neg_one Λ with h2 h2
· simp_all only [ite_true]
· simp_all only [not_true_eq_false]
· rw [if_neg]
· rfl
· cases' det_eq_one_or_neg_one Λ with h2 h2
· rw [h]
linarith
· linarith
lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
detContinuous Λ = detContinuous Λ' ↔ Λ.1.det = Λ'.1.det := by
cases' det_eq_one_or_neg_one Λ with h1 h1
· rw [h1, (detContinuous_eq_one Λ).mpr h1]
cases' det_eq_one_or_neg_one Λ' with h2 h2
· rw [h2, (detContinuous_eq_one Λ').mpr h2]
simp only [toMul_zero]
· rw [h2, (detContinuous_eq_zero Λ').mpr h2]
erw [Additive.toMul.apply_eq_iff_eq]
change (0 : Fin 2) = (1 : Fin 2) ↔ _
simp only [Fin.isValue, zero_ne_one, false_iff]
linarith
· rw [h1, (detContinuous_eq_zero Λ).mpr h1]
cases' det_eq_one_or_neg_one Λ' with h2 h2
· rw [h2, (detContinuous_eq_one Λ').mpr h2]
erw [Additive.toMul.apply_eq_iff_eq]
change (1 : Fin 2) = (0 : Fin 2) ↔ _
simp only [Fin.isValue, one_ne_zero, false_iff]
linarith
· rw [h2, (detContinuous_eq_zero Λ').mpr h2]
simp only [Nat.reduceAdd]
/-- The representation taking a Lorentz matrix to its determinant. -/
@[simps!]
def detRep : 𝓛 d →* ℤ₂ where
toFun Λ := detContinuous Λ
map_one' := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
lorentzGroupIsGroup_one_coe, det_one, coeForℤ₂_apply, ↓reduceIte]
map_mul' Λ1 Λ2 := by
simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul, det_mul, toMul_zero,
mul_ite, mul_one, ite_mul, one_mul]
cases' det_eq_one_or_neg_one Λ1 with h1 h1
· rw [(detContinuous_eq_one Λ1).mpr h1]
cases' det_eq_one_or_neg_one Λ2 with h2 h2
· rw [(detContinuous_eq_one Λ2).mpr h2]
apply (detContinuous_eq_one _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_one]
· rw [(detContinuous_eq_zero Λ2).mpr h2]
apply (detContinuous_eq_zero _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one]
· rw [(detContinuous_eq_zero Λ1).mpr h1]
cases' det_eq_one_or_neg_one Λ2 with h2 h2
· rw [(detContinuous_eq_one Λ2).mpr h2]
apply (detContinuous_eq_zero _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_one]
· rw [(detContinuous_eq_zero Λ2).mpr h2]
apply (detContinuous_eq_one _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one, neg_neg]
lemma detRep_continuous : Continuous (@detRep d) := detContinuous.2
lemma det_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
Λ.1.det = Λ'.1.det := by
obtain ⟨s, hs, hΛ'⟩ := h
let f : ContinuousMap s ℤ₂ := ContinuousMap.restrict s detContinuous
haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
simpa [f, detContinuous_eq_iff_det_eq] using
(@IsPreconnected.subsingleton ℤ₂ _ _ _ (isPreconnected_range f.2))
(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
lemma detRep_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
detRep Λ = detRep Λ' := by
simp only [detRep_apply, detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
coeForℤ₂_apply, Subtype.mk.injEq]
rw [det_on_connected_component h]
| lemma det_of_joined {Λ Λ' : LorentzGroup d} (h : Joined Λ Λ') : Λ.1.det = Λ'.1.det | LorentzGroup.det_of_joined | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "fdf5fda1e73e043ba40a3836ed1f00ddcc259373",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
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"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic"
]
} | {
"hasProof": true,
"proof": ":=\n det_on_connected_component $ pathComponent_subset_component _ h",
"proofType": "term",
"proofLengthLines": 1,
"proofLengthTokens": 68
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
| @[simp]
lemma dual_id : @dual d 1 = 1 | minkowskiMetric.dual_id | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 237,
"tokenPositionInFile": 7406,
"theoremPositionInFile": 22
} | {
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"Lean.Meta.Tactic.Unfold",
"Lean.Elab.Tactic.Unfold",
"Lean.Elab.Tactic.Conv.Unfold",
"Lean.Elab.Tactic.Conv",
"Lean.Elab.Tactic.Meta",
"Lean.Elab.Tactic.Cache",
"Lean.Elab.Calc",
"Lean.Elab.Tactic.Calc",
"Lean.Elab.Tactic.Congr",
"Lean.Elab.Tactic.Guard",
"Lean.Elab.Tactic.Change",
"Lean.Elab.Tactic.FalseOrByContra",
"Lean.Elab.Tactic.Omega.OmegaM",
"Lean.Elab.Tactic.Omega.MinNatAbs",
"Lean.Elab.Tactic.Omega.Core",
"Lean.Elab.Tactic.Omega.Frontend",
"Lean.Elab.Tactic.Omega",
"Lean.Elab.Tactic.Simpa",
"Lean.Meta.Tactic.NormCast",
"Lean.Elab.Tactic.NormCast",
"Lean.Elab.Tactic.Symm",
"Lean.LabelAttribute",
"Lean.Meta.Iterator",
"Lean.Meta.Tactic.IndependentOf",
"Lean.Meta.Tactic.Backtrack",
"Lean.Meta.Tactic.SolveByElim",
"Lean.Elab.Tactic.SolveByElim",
"Lean.Meta.LazyDiscrTree",
"Lean.Util.Heartbeats",
"Lean.Meta.Tactic.LibrarySearch",
"Lean.Elab.Tactic.LibrarySearch",
"Lean.Elab.Tactic.ShowTerm",
"Lean.Elab.Tactic",
"Lean.Elab.StructInst",
"Lean.Elab.Print",
"Lean.Elab.PreDefinition.WF",
"Lean.Elab.PreDefinition",
"Lean.Elab.Deriving.Util",
"Lean.Elab.Deriving.Inhabited",
"Lean.Elab.Deriving.Nonempty",
"Lean.Elab.Deriving.TypeName",
"Lean.Elab.Deriving.BEq",
"Lean.Meta.Inductive",
"Lean.Elab.Deriving.DecEq",
"Lean.Elab.Deriving.Repr",
"Lean.Elab.Deriving.FromToJson",
"Lean.Elab.Deriving.SizeOf",
"Lean.Elab.Deriving.Hashable",
"Lean.Elab.Deriving.Ord",
"Lean.Elab.Deriving",
"Lean.Elab.Extra",
"Lean.Elab.GenInjective",
"Lean.Elab.Macro",
"Lean.Elab.Notation",
"Lean.Elab.Mixfix",
"Lean.Elab.MacroRules",
"Lean.Elab.BuiltinCommand",
"Lean.Elab.InheritDoc",
"Lean.Elab.ParseImportsFast",
"Lean.Elab.GuardMsgs",
"Lean.Elab.CheckTactic",
"Lean.Elab.MatchExpr",
"Lean.Elab",
"Lean.Meta.LevelDefEq",
"Lean.Meta.UnificationHint",
"Lean.Meta.ExprDefEq",
"Lean.Meta.Tactic.LinearArith.Solver",
"Lean.Meta.Tactic.LinearArith.Nat.Solver",
"Lean.Meta.Tactic.LinearArith.Nat",
"Lean.Meta.Tactic.LinearArith.Main",
"Lean.Meta.Tactic.LinearArith",
"Lean.Meta.Tactic.AC.Main",
"Lean.Meta.Tactic.AC",
"Lean.Meta.Tactic",
"Lean.Meta.ExprLens",
"Lean.Meta.ExprTraverse",
"Lean.Meta",
"Lean.Util.ShareCommon",
"Lean.Util.TestExtern",
"Lean.Util.LeanOptions",
"Lean.Util.FileSetupInfo",
"Lean.Util",
"Lean.Server.Watchdog",
"Lean.LoadDynlib",
"Lean.Server.FileWorker.WidgetRequests",
"Lean.Util.LakePath",
"Lean.Server.FileWorker.SetupFile",
"Lean.Server.ImportCompletion",
"Lean.Server.FileWorker",
"Lean.Server.Rpc.Deriving",
"Lean.Server.Rpc",
"Lean.Server",
"Lean.Widget",
"Lean.Linter.Builtin",
"Lean.Linter",
"Lean",
"Std.Classes.BEq",
"Std.Classes.Cast",
"Std.Classes.Order",
"Std.Classes.RatCast",
"Std.Classes.SatisfiesM",
"Std.CodeAction.Misc",
"Std.CodeAction",
"Std.Control.ForInStep.Basic",
"Std.Control.ForInStep.Lemmas",
"Std.Control.ForInStep",
"Std.Control.Lemmas",
"Std.Data.MLList.Basic",
"Std.Control.Nondet.Basic",
"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
"Std.Data.Bool",
"Std.Data.Fin.Basic",
"Std.Data.Option.Lemmas",
"Std.Data.List.Lemmas",
"Std.Tactic.SeqFocus",
"Std.Util.ProofWanted",
"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
"Std.Data.BinomialHeap.Basic",
"Std.Data.BinomialHeap.Lemmas",
"Std.Data.BinomialHeap",
"Std.Data.Fin.Lemmas",
"Std.Data.BitVec.Lemmas",
"Std.Data.BitVec",
"Std.Data.ByteArray",
"Std.Data.Char",
"Std.Data.DList",
"Std.Data.Fin",
"Std.Data.HashMap.Basic",
"Std.Data.HashMap.Lemmas",
"Std.Data.HashMap.WF",
"Std.Data.HashMap",
"Std.Data.Int.Gcd",
"Std.Data.Int.Lemmas",
"Std.Data.Int",
"Std.Data.LazyList",
"Std.Data.List.Count",
"Std.Data.List.Pairwise",
"Std.Data.List.Perm",
"Std.Data.List",
"Std.Data.MLList.Heartbeats",
"Std.Lean.System.IO",
"Std.Data.MLList.IO",
"Std.Data.MLList",
"Std.Data.Nat",
"Std.Data.Option",
"Std.Data.PairingHeap",
"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
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} | {
"hasProof": true,
"proof": ":= by\n simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq",
"proofType": "tactic",
"proofLengthLines": 1,
"proofLengthTokens": 74
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
import HepLean.Mathematics.SO3.Basic
import Mathlib.Topology.Constructions
/-!
# Rotations
This file describes the embedding of `SO(3)` into `LorentzGroup 3`.
-/
/-! TODO: Generalize the inclusion of rotations into LorentzGroup to abitary dimension. -/
noncomputable section
namespace LorentzGroup
open GroupTheory
/-- Given a element of `SO(3)` the matrix corresponding to a space-time rotation. -/
@[simp]
def SO3ToMatrix (A : SO(3)) : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ :=
(Matrix.fromBlocks 1 0 0 A.1)
| lemma SO3ToMatrix_in_LorentzGroup (A : SO(3)) : SO3ToMatrix A ∈ LorentzGroup 3 | LorentzGroup.SO3ToMatrix_in_LorentzGroup | {
"commit": "026ed8b85e7a3c83656d9de99ef81bf41ac20e9e",
"date": "2024-05-21T00:00:00"
} | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Rotations.lean | HepLean.SpaceTime.LorentzGroup.Rotations | HepLean.SpaceTime.LorentzGroup.Rotations.jsonl | {
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"tokenPositionInFile": 725,
"theoremPositionInFile": 1
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"Lean.Elab.GuardMsgs",
"Lean.Elab.CheckTactic",
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"Lean.Meta.Tactic.LinearArith.Solver",
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"Lean.Meta.Tactic.LinearArith",
"Lean.Meta.Tactic.AC.Main",
"Lean.Meta.Tactic.AC",
"Lean.Meta.Tactic",
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"Lean.Meta",
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"Lean.Util",
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"Lean.Util.LakePath",
"Lean.Server.FileWorker.SetupFile",
"Lean.Server.ImportCompletion",
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"Lean.Server.Rpc.Deriving",
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"Lean.Server",
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"Lean",
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"Std.Control.Lemmas",
"Std.Data.MLList.Basic",
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"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
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"Std.Data.List.Lemmas",
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"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
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"Std.Data.BinomialHeap",
"Std.Data.Fin.Lemmas",
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"Std.Data.List",
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"Std.Data.MLList.IO",
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"Std.Data.RBMap.Basic",
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"Std.Data.RBMap.Alter",
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"Std.Data.Range.Lemmas",
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"Std.Data.Rat.Lemmas",
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"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
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"Std.Lean.Delaborator",
"Std.Lean.Except",
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"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
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"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
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"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
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"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
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"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
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"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
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"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
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"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
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"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
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"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
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"Mathlib.Algebra.DirectSum.Module",
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} | {
"hasProof": true,
"proof": ":= by\n rw [LorentzGroup.mem_iff_dual_mul_self]\n simp only [minkowskiMetric.dual, minkowskiMatrix.as_block, SO3ToMatrix,\n Matrix.fromBlocks_transpose, Matrix.transpose_one, Matrix.transpose_zero,\n Matrix.fromBlocks_multiply, mul_one, Matrix.mul_zero, add_zero, Matrix.zero_mul, Matrix.mul_one,\n neg_mul, one_mul, zero_add, Matrix.mul_neg, neg_zero, mul_neg, neg_neg,\n Matrix.mul_eq_one_comm.mpr A.2.2, Matrix.fromBlocks_one]",
"proofType": "tactic",
"proofLengthLines": 6,
"proofLengthTokens": 437
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space time
This file introduce 4d Minkowski spacetime.
-/
noncomputable section
/-- The space-time -/
def SpaceTime : Type := Fin 4 → ℝ
/-- Give spacetime the structure of an additive commutative monoid. -/
instance : AddCommMonoid SpaceTime := Pi.addCommMonoid
/-- Give spacetime the structure of a module over the reals. -/
instance : Module ℝ SpaceTime := Pi.module _ _ _
instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normedAddCommGroup
instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
namespace SpaceTime
open Manifold
open Matrix
open Complex
open ComplexConjugate
/-- The space part of spacetime. -/
@[simp]
def space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
/-- The structure of a smooth manifold on spacetime. -/
def asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime := 𝓘(ℝ, SpaceTime)
instance : ChartedSpace SpaceTime SpaceTime := chartedSpaceSelf SpaceTime
/-- The standard basis for spacetime. -/
def stdBasis : Basis (Fin 4) ℝ SpaceTime := Pi.basisFun ℝ (Fin 4)
lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 := by
erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']
lemma stdBasis_not_eq {μ ν : Fin 4} (h : μ ≠ ν) : stdBasis μ ν = 0 := by
rw [stdBasis_apply]
exact if_neg h
lemma stdBasis_0 : stdBasis 0 = ![1, 0, 0, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_1 : stdBasis 1 = ![0, 1, 0, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_2 : stdBasis 2 = ![0, 0, 1, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
| lemma stdBasis_3 : stdBasis 3 = ![0, 0, 0, 1] | SpaceTime.stdBasis_3 | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "db0f6de59c636e635219f2013bf9cad19a2d09ca",
"date": "2024-05-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
"lineInFile": 68,
"tokenPositionInFile": 1956,
"theoremPositionInFile": 9
} | {
"inFilePremises": true,
"numInFilePremises": 5,
"repositoryPremises": true,
"numRepositoryPremises": 5,
"numPremises": 68,
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"Lean.Meta.Tactic.LinearArith",
"Lean.Meta.Tactic.AC.Main",
"Lean.Meta.Tactic.AC",
"Lean.Meta.Tactic",
"Lean.Meta.ExprLens",
"Lean.Meta.ExprTraverse",
"Lean.Meta",
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"Lean.Util.TestExtern",
"Lean.Util.LeanOptions",
"Lean.Util.FileSetupInfo",
"Lean.Util",
"Lean.Server.Watchdog",
"Lean.LoadDynlib",
"Lean.Server.FileWorker.WidgetRequests",
"Lean.Util.LakePath",
"Lean.Server.FileWorker.SetupFile",
"Lean.Server.ImportCompletion",
"Lean.Server.FileWorker",
"Lean.Server.Rpc.Deriving",
"Lean.Server.Rpc",
"Lean.Server",
"Lean.Widget",
"Lean.Linter.Builtin",
"Lean.Linter",
"Lean",
"Std.Classes.BEq",
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"Std.Classes.RatCast",
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"Std.Control.ForInStep.Basic",
"Std.Control.ForInStep.Lemmas",
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"Std.Control.Lemmas",
"Std.Data.MLList.Basic",
"Std.Control.Nondet.Basic",
"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
"Std.Data.Bool",
"Std.Data.Fin.Basic",
"Std.Data.Option.Lemmas",
"Std.Data.List.Lemmas",
"Std.Tactic.SeqFocus",
"Std.Util.ProofWanted",
"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
"Std.Data.BinomialHeap.Basic",
"Std.Data.BinomialHeap.Lemmas",
"Std.Data.BinomialHeap",
"Std.Data.Fin.Lemmas",
"Std.Data.BitVec.Lemmas",
"Std.Data.BitVec",
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"Std.Data.HashMap.Lemmas",
"Std.Data.HashMap.WF",
"Std.Data.HashMap",
"Std.Data.Int.Gcd",
"Std.Data.Int.Lemmas",
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"Std.Data.List.Count",
"Std.Data.List.Pairwise",
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"Std.Data.List",
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"Std.Lean.System.IO",
"Std.Data.MLList.IO",
"Std.Data.MLList",
"Std.Data.Nat",
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"Std.Data.PairingHeap",
"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Hull",
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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
lemma IsOrthochronous_iff_ge_one :
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
NormOneLorentzVector.time_pos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_neg_one :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ -1 := by
rw [timeComp, IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.not_mem_iff,
NormOneLorentzVector.time_nonpos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_zero :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [not_orthochronous_iff_le_neg_one] at h
linarith
· rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
def stepFunction : ℝ → ℝ := fun t =>
if t ≤ -1 then -1 else
if 1 ≤ t then 1 else t
lemma stepFunction_continuous : Continuous stepFunction := by
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
<;> intro a ha
· rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp only [le_neg_self_iff, id_eq]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
· rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
exact id (Eq.symm ha)
/-- The continuous map from `lorentzGroup` to `ℝ` wh
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
def orthchroMapReal : C(LorentzGroup d, ℝ) := ContinuousMap.comp
⟨stepFunction, stepFunction_continuous⟩ timeCompCont
lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMapReal Λ = 1 := by
rw [IsOrthochronous_iff_ge_one, timeComp] at h
change stepFunction (Λ.1 _ _) = 1
rw [stepFunction, if_pos h, if_neg]
linarith
| lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMapReal Λ = - 1 | LorentzGroup.orthchroMapReal_on_not_IsOrthochronous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"Std.Lean.Meta.Expr",
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"Std.Lean.MonadBacktrack",
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"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
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"Std.Tactic.Unreachable",
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"Std",
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"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
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"Mathlib.Algebra.BigOperators.List.Defs",
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"Mathlib.Init.Data.Nat.Basic",
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"Mathlib.Tactic.ProjectionNotation",
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"Mathlib.Tactic.Core",
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"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
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"Aesop.Frontend.Extension",
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"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
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"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
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"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
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"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
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"Mathlib.Algebra.Order.Group.OrderIso",
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"Mathlib.Algebra.BigOperators.Multiset.Order",
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"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
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"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
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"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
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"Mathlib.Algebra.Order.Monoid.WithTop",
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"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
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"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
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"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
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"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"HepLean.SpaceTime.LorentzGroup.Proper"
]
} | {
"hasProof": true,
"proof": ":= by\n rw [not_orthochronous_iff_le_neg_one] at h\n change stepFunction (timeComp _)= - 1\n rw [stepFunction, if_pos h]",
"proofType": "tactic",
"proofLengthLines": 3,
"proofLengthTokens": 120
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
simp only [unitUp, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
/-- The unit of the contraction of covariant Lorentz vector and a
contravariant Lorentz vector. -/
def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x := by
simp only [unitDown, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,
lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']
/-!
# Contractions and the Lorentz actions
-/
open Matrix
@[simp]
lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) := by
rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
@[simp]
lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
(LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) := by
rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]
rw [dotProduct_comm, dotProduct_comm x y]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
end LorentzVector
namespace minkowskiMatrix
open LorentzVector
open TensorStructure
open scoped minkowskiMatrix
variable {d : ℕ}
/-- The metric tensor as an element of `LorentzVector d ⊗[ℝ] LorentzVector d`. -/
def asTenProd : LorentzVector d ⊗[ℝ] LorentzVector d :=
∑ μ, ∑ ν, η μ ν • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis ν)
| lemma asTenProd_diag :
@asTenProd d = ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ) | minkowskiMatrix.asTenProd_diag | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 181,
"tokenPositionInFile": 6842,
"theoremPositionInFile": 13
} | {
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"ProofWidgets.Component.Basic",
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"ProofWidgets.Data.Html",
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"ProofWidgets.Component.OfRpcMethod",
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"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
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"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
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"Mathlib.Topology.UnitInterval",
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"Mathlib.Order.Copy",
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"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
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"HepLean.SpaceTime.LorentzGroup.Basic",
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"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic",
"HepLean.SpaceTime.LorentzVector.LorentzAction",
"HepLean.SpaceTime.LorentzVector.Covariant",
"Mathlib.LinearAlgebra.Multilinear.TensorProduct",
"Mathlib.LinearAlgebra.PiTensorProduct",
"HepLean.Tensors.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [asTenProd]\n refine Finset.sum_congr rfl (fun μ _ => ?_)\n rw [Finset.sum_eq_single μ]\n · intro ν _ hμν\n rw [minkowskiMatrix.off_diag_zero hμν.symm]\n exact TensorProduct.zero_smul (e μ ⊗ₜ[ℝ] e ν)\n · intro a\n rename_i j\n exact False.elim (a j)",
"proofType": "tactic",
"proofLengthLines": 9,
"proofLengthTokens": 275
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
simp only [unitUp, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
/-- The unit of the contraction of covariant Lorentz vector and a
contravariant Lorentz vector. -/
def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
| lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x | LorentzVector.unitDown_rid | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a97cb623798a037cb7f9ff27508a67bec560f6c7",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 135,
"tokenPositionInFile": 4937,
"theoremPositionInFile": 9
} | {
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"numInFilePremises": 4,
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"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
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"Mathlib.Data.Nat.Cast.Prod",
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"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
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"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
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"Mathlib.Algebra.Module.Submodule.LinearMap",
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"Mathlib.Order.Hom.CompleteLattice",
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"Mathlib.Data.List.NatAntidiagonal",
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"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
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"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
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"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
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"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
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"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
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"Mathlib.Data.List.FinRange",
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"Mathlib.Data.Finsupp.Fin",
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"Mathlib.Algebra.Algebra.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
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"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
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"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
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"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.LinearAlgebra.Pi",
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"Mathlib.Algebra.Module.Submodule.Pointwise",
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"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
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"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
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"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
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"Mathlib.Algebra.Order.Pi",
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"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
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"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
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"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
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"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
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"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
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"Mathlib.Topology.Order.MonotoneContinuity",
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"Mathlib.Data.Complex.Abs",
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"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
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"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
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"Mathlib.GroupTheory.Submonoid.Order",
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"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
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"Mathlib.Data.Real.Cardinality",
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"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
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"Mathlib.Algebra.Algebra.Pi",
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"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
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"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
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"Mathlib.Init.Data.Prod",
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"Mathlib.Algebra.FreeAlgebra",
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"Mathlib.Data.Nat.Count",
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"Mathlib.GroupTheory.Subgroup.Simple",
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"Mathlib.GroupTheory.PGroup",
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"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
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"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
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"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
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"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
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"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
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"Mathlib.Tactic.ComputeDegree",
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"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
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"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
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"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
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"Mathlib.Analysis.Convex.Topology",
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"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
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"Mathlib.Algebra.BigOperators.Module",
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"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
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"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic",
"HepLean.SpaceTime.LorentzVector.LorentzAction",
"HepLean.SpaceTime.LorentzVector.Covariant",
"Mathlib.LinearAlgebra.Multilinear.TensorProduct",
"Mathlib.LinearAlgebra.PiTensorProduct",
"HepLean.Tensors.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [unitDown, LinearEquiv.refl_toLinearMap]\n rw [tmul_sum]\n simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,\n Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,\n Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,\n assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,\n lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']",
"proofType": "tactic",
"proofLengthLines": 7,
"proofLengthTokens": 496
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`.
We define the give a series of lemmas related to the determinant of the Lorentz group.
-/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
open minkowskiMetric
variable {d : ℕ}
/-- The determinant of a member of the Lorentz group is `1` or `-1`. -/
lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
refine mul_self_eq_one_iff.mp ?_
simpa only [det_mul, det_dual, det_one] using congrArg det ((mem_iff_self_mul_dual).mp Λ.2)
local notation "ℤ₂" => Multiplicative (ZMod 2)
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
instance : DiscreteTopology ℤ₂ := by
exact forall_open_iff_discrete.mp fun _ => trivial
instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
instance _root_.Finite.instDiscreteTopology [TopologicalSpace X] [T1Space X] [Finite X] : DiscreteTopology X :=
discreteTopology_iff_forall_isClosed.mpr (· |>.toFinite.isClosed)
/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
@[simps!]
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
toFun x := if x = ⟨1, Set.mem_insert_of_mem (-1) rfl⟩
then Additive.toMul 0 else Additive.toMul (1 : ZMod 2)
continuous_toFun := continuous_of_discreteTopology
/-- The continuous map taking a Lorentz matrix to its determinant. -/
def detContinuous : C(𝓛 d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨Λ.1.det, Or.symm (LorentzGroup.det_eq_one_or_neg_one _)⟩,
continuous_toFun := by
refine Continuous.subtype_mk ?_ _
exact Continuous.matrix_det $
Continuous.comp' (continuous_iff_le_induced.mpr fun U a => a) continuous_id'
}
@[simp]
lemma _root_.toMul_eq_one {α : Type*} [One α] {x : Additive α} :
Additive.toMul x = 1 ↔ x = 0 :=
Iff.rfl
lemma detContinuous_eq_one (Λ : LorentzGroup d) :
detContinuous Λ = Additive.toMul 0 ↔ Λ.1.det = 1 := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
Subtype.mk.injEq, ite_eq_left_iff, toMul_eq_one]
simp only [toMul_zero, ite_eq_left_iff, toMul_eq_one]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· by_contra hn
have h' := h hn
change (1 : Fin 2) = (0 : Fin 2) at h'
simp only [Fin.isValue, one_ne_zero] at h'
· intro h'
exact False.elim (h' h)
lemma detContinuous_eq_zero (Λ : LorentzGroup d) :
detContinuous Λ = Additive.toMul (1 : ZMod 2) ↔ Λ.1.det = - 1 := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
Subtype.mk.injEq, Nat.reduceAdd]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· by_contra hn
rw [if_pos] at h
· change (0 : Fin 2) = (1 : Fin 2) at h
simp only [Fin.isValue, zero_ne_one] at h
· cases' det_eq_one_or_neg_one Λ with h2 h2
· simp_all only [ite_true]
· simp_all only [not_true_eq_false]
· rw [if_neg]
· rfl
· cases' det_eq_one_or_neg_one Λ with h2 h2
· rw [h]
linarith
· linarith
lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
detContinuous Λ = detContinuous Λ' ↔ Λ.1.det = Λ'.1.det := by
cases' det_eq_one_or_neg_one Λ with h1 h1
· rw [h1, (detContinuous_eq_one Λ).mpr h1]
cases' det_eq_one_or_neg_one Λ' with h2 h2
· rw [h2, (detContinuous_eq_one Λ').mpr h2]
simp only [toMul_zero]
· rw [h2, (detContinuous_eq_zero Λ').mpr h2]
erw [Additive.toMul.apply_eq_iff_eq]
change (0 : Fin 2) = (1 : Fin 2) ↔ _
simp only [Fin.isValue, zero_ne_one, false_iff]
linarith
· rw [h1, (detContinuous_eq_zero Λ).mpr h1]
cases' det_eq_one_or_neg_one Λ' with h2 h2
· rw [h2, (detContinuous_eq_one Λ').mpr h2]
erw [Additive.toMul.apply_eq_iff_eq]
change (1 : Fin 2) = (0 : Fin 2) ↔ _
simp only [Fin.isValue, one_ne_zero, false_iff]
linarith
· rw [h2, (detContinuous_eq_zero Λ').mpr h2]
simp only [Nat.reduceAdd]
/-- The representation taking a Lorentz matrix to its determinant. -/
@[simps!]
def detRep : 𝓛 d →* ℤ₂ where
toFun Λ := detContinuous Λ
map_one' := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
lorentzGroupIsGroup_one_coe, det_one, coeForℤ₂_apply, ↓reduceIte]
map_mul' Λ1 Λ2 := by
simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul, det_mul, toMul_zero,
mul_ite, mul_one, ite_mul, one_mul]
cases' det_eq_one_or_neg_one Λ1 with h1 h1
· rw [(detContinuous_eq_one Λ1).mpr h1]
cases' det_eq_one_or_neg_one Λ2 with h2 h2
· rw [(detContinuous_eq_one Λ2).mpr h2]
apply (detContinuous_eq_one _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_one]
· rw [(detContinuous_eq_zero Λ2).mpr h2]
apply (detContinuous_eq_zero _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one]
· rw [(detContinuous_eq_zero Λ1).mpr h1]
cases' det_eq_one_or_neg_one Λ2 with h2 h2
· rw [(detContinuous_eq_one Λ2).mpr h2]
apply (detContinuous_eq_zero _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_one]
· rw [(detContinuous_eq_zero Λ2).mpr h2]
apply (detContinuous_eq_one _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one, neg_neg]
lemma detRep_continuous : Continuous (@detRep d) := detContinuous.2
lemma det_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
Λ.1.det = Λ'.1.det := by
obtain ⟨s, hs, hΛ'⟩ := h
let f : ContinuousMap s ℤ₂ := ContinuousMap.restrict s detContinuous
haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
simpa [f, detContinuous_eq_iff_det_eq] using
(@IsPreconnected.subsingleton ℤ₂ _ _ _ (isPreconnected_range f.2))
(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
lemma detRep_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
detRep Λ = detRep Λ' := by
simp only [detRep_apply, detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
coeForℤ₂_apply, Subtype.mk.injEq]
rw [det_on_connected_component h]
lemma det_of_joined {Λ Λ' : LorentzGroup d} (h : Joined Λ Λ') : Λ.1.det = Λ'.1.det :=
det_on_connected_component $ pathComponent_subset_component _ h
/-- A Lorentz Matrix is proper if its determinant is 1. -/
@[simp]
def IsProper (Λ : LorentzGroup d) : Prop := Λ.1.det = 1
instance : DecidablePred (@IsProper d) := by
intro Λ
apply Real.decidableEq
lemma IsProper_iff (Λ : LorentzGroup d) : IsProper Λ ↔ detRep Λ = 1 := by
rw [show 1 = detRep 1 from Eq.symm (MonoidHom.map_one detRep), detRep_apply, detRep_apply,
detContinuous_eq_iff_det_eq]
simp only [IsProper, lorentzGroupIsGroup_one_coe, det_one]
| lemma id_IsProper : @IsProper d 1 | LorentzGroup.id_IsProper | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "8024aa92a17078b8ce8608d256f65d2d82aecb41",
"date": "2024-05-20T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
"lineInFile": 182,
"tokenPositionInFile": 7048,
"theoremPositionInFile": 12
} | {
"inFilePremises": true,
"numInFilePremises": 1,
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"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
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"Mathlib.Algebra.CharP.Reduced",
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"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp [IsProper]",
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} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
@[simp]
lemma dual_id : @dual d 1 = 1 := by
simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
@[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
simp only [dual, transpose_mul]
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
· noncomm_ring [minkowskiMatrix.sq]
· noncomm_ring
@[simp]
lemma dual_dual : dual (dual Λ) = Λ := by
simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
trans (η * η) * Λ * (η * η)
· noncomm_ring
· noncomm_ring [minkowskiMatrix.sq]
| @[simp]
lemma dual_eta : @dual d η = η | minkowskiMetric.dual_eta | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Std.Control.Nondet.Basic",
"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
"Std.Data.Bool",
"Std.Data.Fin.Basic",
"Std.Data.Option.Lemmas",
"Std.Data.List.Lemmas",
"Std.Tactic.SeqFocus",
"Std.Util.ProofWanted",
"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
"Std.Data.BinomialHeap.Basic",
"Std.Data.BinomialHeap.Lemmas",
"Std.Data.BinomialHeap",
"Std.Data.Fin.Lemmas",
"Std.Data.BitVec.Lemmas",
"Std.Data.BitVec",
"Std.Data.ByteArray",
"Std.Data.Char",
"Std.Data.DList",
"Std.Data.Fin",
"Std.Data.HashMap.Basic",
"Std.Data.HashMap.Lemmas",
"Std.Data.HashMap.WF",
"Std.Data.HashMap",
"Std.Data.Int.Gcd",
"Std.Data.Int.Lemmas",
"Std.Data.Int",
"Std.Data.LazyList",
"Std.Data.List.Count",
"Std.Data.List.Pairwise",
"Std.Data.List.Perm",
"Std.Data.List",
"Std.Data.MLList.Heartbeats",
"Std.Lean.System.IO",
"Std.Data.MLList.IO",
"Std.Data.MLList",
"Std.Data.Nat",
"Std.Data.Option",
"Std.Data.PairingHeap",
"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
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]
} | {
"hasProof": true,
"proof": ":= by\n simp only [dual, eq_transpose]\n noncomm_ring [minkowskiMatrix.sq]",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 74
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
lemma IsOrthochronous_iff_ge_one :
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
NormOneLorentzVector.time_pos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_neg_one :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ -1 := by
rw [timeComp, IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.not_mem_iff,
NormOneLorentzVector.time_nonpos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_zero :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [not_orthochronous_iff_le_neg_one] at h
linarith
· rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
def stepFunction : ℝ → ℝ := fun t =>
if t ≤ -1 then -1 else
if 1 ≤ t then 1 else t
lemma stepFunction_continuous : Continuous stepFunction := by
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
<;> intro a ha
· rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp only [le_neg_self_iff, id_eq]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
· rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
exact id (Eq.symm ha)
/-- The continuous map from `lorentzGroup` to `ℝ` wh
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
def orthchroMapReal : C(LorentzGroup d, ℝ) := ContinuousMap.comp
⟨stepFunction, stepFunction_continuous⟩ timeCompCont
lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMapReal Λ = 1 := by
rw [IsOrthochronous_iff_ge_one, timeComp] at h
change stepFunction (Λ.1 _ _) = 1
rw [stepFunction, if_pos h, if_neg]
linarith
lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMapReal Λ = - 1 := by
rw [not_orthochronous_iff_le_neg_one] at h
change stepFunction (timeComp _)= - 1
rw [stepFunction, if_pos h]
lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
by_cases h : IsOrthochronous Λ
· exact Or.inr $ orthchroMapReal_on_IsOrthochronous h
· exact Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
local notation "ℤ₂" => Multiplicative (ZMod 2)
/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
def orthchroMap : C(LorentzGroup d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMap Λ = 1 := by
simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
simp only [orthchroMap, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
orthchroMapReal_on_not_IsOrthochronous h, coeForℤ₂_apply, Subtype.mk.injEq, Nat.reduceAdd]
rw [if_neg]
· rfl
· linarith
lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
(h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
rw [IsOrthochronous_iff_transpose] at h
rw [IsOrthochronous_iff_futurePointing] at h h'
rw [IsOrthochronous, timeComp_mul]
exact NormOneLorentzVector.FuturePointing.metric_reflect_mem_mem h h'
| lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') | LorentzGroup.mul_othchron_of_not_othchron_not_othchron | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"Lean.Meta.Tactic.NormCast",
"Lean.Elab.Tactic.NormCast",
"Lean.Elab.Tactic.Symm",
"Lean.LabelAttribute",
"Lean.Meta.Iterator",
"Lean.Meta.Tactic.IndependentOf",
"Lean.Meta.Tactic.Backtrack",
"Lean.Meta.Tactic.SolveByElim",
"Lean.Elab.Tactic.SolveByElim",
"Lean.Meta.LazyDiscrTree",
"Lean.Util.Heartbeats",
"Lean.Meta.Tactic.LibrarySearch",
"Lean.Elab.Tactic.LibrarySearch",
"Lean.Elab.Tactic.ShowTerm",
"Lean.Elab.Tactic",
"Lean.Elab.StructInst",
"Lean.Elab.Print",
"Lean.Elab.PreDefinition.WF",
"Lean.Elab.PreDefinition",
"Lean.Elab.Deriving.Util",
"Lean.Elab.Deriving.Inhabited",
"Lean.Elab.Deriving.Nonempty",
"Lean.Elab.Deriving.TypeName",
"Lean.Elab.Deriving.BEq",
"Lean.Meta.Inductive",
"Lean.Elab.Deriving.DecEq",
"Lean.Elab.Deriving.Repr",
"Lean.Elab.Deriving.FromToJson",
"Lean.Elab.Deriving.SizeOf",
"Lean.Elab.Deriving.Hashable",
"Lean.Elab.Deriving.Ord",
"Lean.Elab.Deriving",
"Lean.Elab.Extra",
"Lean.Elab.GenInjective",
"Lean.Elab.Macro",
"Lean.Elab.Notation",
"Lean.Elab.Mixfix",
"Lean.Elab.MacroRules",
"Lean.Elab.BuiltinCommand",
"Lean.Elab.InheritDoc",
"Lean.Elab.ParseImportsFast",
"Lean.Elab.GuardMsgs",
"Lean.Elab.CheckTactic",
"Lean.Elab.MatchExpr",
"Lean.Elab",
"Lean.Meta.LevelDefEq",
"Lean.Meta.UnificationHint",
"Lean.Meta.ExprDefEq",
"Lean.Meta.Tactic.LinearArith.Solver",
"Lean.Meta.Tactic.LinearArith.Nat.Solver",
"Lean.Meta.Tactic.LinearArith.Nat",
"Lean.Meta.Tactic.LinearArith.Main",
"Lean.Meta.Tactic.LinearArith",
"Lean.Meta.Tactic.AC.Main",
"Lean.Meta.Tactic.AC",
"Lean.Meta.Tactic",
"Lean.Meta.ExprLens",
"Lean.Meta.ExprTraverse",
"Lean.Meta",
"Lean.Util.ShareCommon",
"Lean.Util.TestExtern",
"Lean.Util.LeanOptions",
"Lean.Util.FileSetupInfo",
"Lean.Util",
"Lean.Server.Watchdog",
"Lean.LoadDynlib",
"Lean.Server.FileWorker.WidgetRequests",
"Lean.Util.LakePath",
"Lean.Server.FileWorker.SetupFile",
"Lean.Server.ImportCompletion",
"Lean.Server.FileWorker",
"Lean.Server.Rpc.Deriving",
"Lean.Server.Rpc",
"Lean.Server",
"Lean.Widget",
"Lean.Linter.Builtin",
"Lean.Linter",
"Lean",
"Std.Classes.BEq",
"Std.Classes.Cast",
"Std.Classes.Order",
"Std.Classes.RatCast",
"Std.Classes.SatisfiesM",
"Std.CodeAction.Misc",
"Std.CodeAction",
"Std.Control.ForInStep.Basic",
"Std.Control.ForInStep.Lemmas",
"Std.Control.ForInStep",
"Std.Control.Lemmas",
"Std.Data.MLList.Basic",
"Std.Control.Nondet.Basic",
"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
"Std.Data.Bool",
"Std.Data.Fin.Basic",
"Std.Data.Option.Lemmas",
"Std.Data.List.Lemmas",
"Std.Tactic.SeqFocus",
"Std.Util.ProofWanted",
"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
"Std.Data.BinomialHeap.Basic",
"Std.Data.BinomialHeap.Lemmas",
"Std.Data.BinomialHeap",
"Std.Data.Fin.Lemmas",
"Std.Data.BitVec.Lemmas",
"Std.Data.BitVec",
"Std.Data.ByteArray",
"Std.Data.Char",
"Std.Data.DList",
"Std.Data.Fin",
"Std.Data.HashMap.Basic",
"Std.Data.HashMap.Lemmas",
"Std.Data.HashMap.WF",
"Std.Data.HashMap",
"Std.Data.Int.Gcd",
"Std.Data.Int.Lemmas",
"Std.Data.Int",
"Std.Data.LazyList",
"Std.Data.List.Count",
"Std.Data.List.Pairwise",
"Std.Data.List.Perm",
"Std.Data.List",
"Std.Data.MLList.Heartbeats",
"Std.Lean.System.IO",
"Std.Data.MLList.IO",
"Std.Data.MLList",
"Std.Data.Nat",
"Std.Data.Option",
"Std.Data.PairingHeap",
"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"HepLean.SpaceTime.LorentzGroup.Proper"
]
} | {
"hasProof": true,
"proof": ":= by\n rw [IsOrthochronous_iff_transpose] at h\n rw [IsOrthochronous_iff_futurePointing] at h h'\n rw [IsOrthochronous, timeComp_mul]\n exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem h h'",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 214
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
| lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x | LorentzVector.unitUp_rid | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a97cb623798a037cb7f9ff27508a67bec560f6c7",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 121,
"tokenPositionInFile": 4148,
"theoremPositionInFile": 7
} | {
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"Mathlib.Tactic.SudoSetOption",
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"ProofWidgets.Component.Basic",
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"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
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"Mathlib.Util.CountHeartbeats",
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"Mathlib.Data.FunLike.Basic",
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"Mathlib.Data.Nat.Cast.Defs",
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"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic",
"HepLean.SpaceTime.LorentzVector.LorentzAction",
"HepLean.SpaceTime.LorentzVector.Covariant",
"Mathlib.LinearAlgebra.Multilinear.TensorProduct",
"Mathlib.LinearAlgebra.PiTensorProduct",
"HepLean.Tensors.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [unitUp, LinearEquiv.refl_toLinearMap]\n rw [tmul_sum]\n simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,\n Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,\n LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,\n contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']",
"proofType": "tactic",
"proofLengthLines": 6,
"proofLengthTokens": 442
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`.
We define the give a series of lemmas related to the determinant of the Lorentz group.
-/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
open minkowskiMetric
variable {d : ℕ}
/-- The determinant of a member of the Lorentz group is `1` or `-1`. -/
lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
refine mul_self_eq_one_iff.mp ?_
simpa only [det_mul, det_dual, det_one] using congrArg det ((mem_iff_self_mul_dual).mp Λ.2)
local notation "ℤ₂" => Multiplicative (ZMod 2)
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
instance : DiscreteTopology ℤ₂ := by
exact forall_open_iff_discrete.mp fun _ => trivial
instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
instance _root_.Finite.instDiscreteTopology [TopologicalSpace X] [T1Space X] [Finite X] : DiscreteTopology X :=
discreteTopology_iff_forall_isClosed.mpr (· |>.toFinite.isClosed)
/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
@[simps!]
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
toFun x := if x = ⟨1, Set.mem_insert_of_mem (-1) rfl⟩
then Additive.toMul 0 else Additive.toMul (1 : ZMod 2)
continuous_toFun := continuous_of_discreteTopology
/-- The continuous map taking a Lorentz matrix to its determinant. -/
def detContinuous : C(𝓛 d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨Λ.1.det, Or.symm (LorentzGroup.det_eq_one_or_neg_one _)⟩,
continuous_toFun := by
refine Continuous.subtype_mk ?_ _
exact Continuous.matrix_det $
Continuous.comp' (continuous_iff_le_induced.mpr fun U a => a) continuous_id'
}
@[simp]
lemma _root_.toMul_eq_one {α : Type*} [One α] {x : Additive α} :
Additive.toMul x = 1 ↔ x = 0 :=
Iff.rfl
lemma detContinuous_eq_one (Λ : LorentzGroup d) :
detContinuous Λ = Additive.toMul 0 ↔ Λ.1.det = 1 := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
Subtype.mk.injEq, ite_eq_left_iff, toMul_eq_one]
simp only [toMul_zero, ite_eq_left_iff, toMul_eq_one]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· by_contra hn
have h' := h hn
change (1 : Fin 2) = (0 : Fin 2) at h'
simp only [Fin.isValue, one_ne_zero] at h'
· intro h'
exact False.elim (h' h)
lemma detContinuous_eq_zero (Λ : LorentzGroup d) :
detContinuous Λ = Additive.toMul (1 : ZMod 2) ↔ Λ.1.det = - 1 := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
Subtype.mk.injEq, Nat.reduceAdd]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· by_contra hn
rw [if_pos] at h
· change (0 : Fin 2) = (1 : Fin 2) at h
simp only [Fin.isValue, zero_ne_one] at h
· cases' det_eq_one_or_neg_one Λ with h2 h2
· simp_all only [ite_true]
· simp_all only [not_true_eq_false]
· rw [if_neg]
· rfl
· cases' det_eq_one_or_neg_one Λ with h2 h2
· rw [h]
linarith
· linarith
| lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
detContinuous Λ = detContinuous Λ' ↔ Λ.1.det = Λ'.1.det | LorentzGroup.detContinuous_eq_iff_det_eq | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
"lineInFile": 99,
"tokenPositionInFile": 3283,
"theoremPositionInFile": 5
} | {
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"ProofWidgets.Component.Basic",
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"ProofWidgets.Data.Html",
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"ProofWidgets.Component.OfRpcMethod",
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"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
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"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
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"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
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"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
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"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
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"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
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"Mathlib.Topology.Algebra.Order.Compact",
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"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
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"Mathlib.Algebra.Order.Module.Pointwise",
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"Mathlib.Data.Real.Pointwise",
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"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
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"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
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"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
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"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
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"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
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"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
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"Mathlib.Data.Matrix.Basic",
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"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
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"Mathlib.LinearAlgebra.Dimension.Basic",
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"Mathlib.RingTheory.Congruence",
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"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
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"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Algebra.Ring.Fin",
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"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Data.Fintype.Perm",
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"Mathlib.GroupTheory.Perm.Finite",
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"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
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"Mathlib.Init.Data.Prod",
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"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
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"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
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"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
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"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
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"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
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"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
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"Mathlib.Data.Polynomial.Div",
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"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
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"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
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"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
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"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
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"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
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"Mathlib.FieldTheory.Tower",
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"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
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"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
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"Mathlib.Analysis.Normed.Group.Basic",
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"Mathlib.RingTheory.NonUnitalSubring.Basic",
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"Mathlib.Topology.MetricSpace.Dilation",
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"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
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"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
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"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
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"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
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"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
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"Mathlib.LinearAlgebra.Matrix.Basis",
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"Mathlib.Topology.Instances.Matrix",
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"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
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"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
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"Mathlib.Analysis.Normed.Group.AddCircle",
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"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
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"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
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"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
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"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
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]
} | {
"hasProof": true,
"proof": ":= by\n cases' det_eq_one_or_neg_one Λ with h1 h1\n · rw [h1, (detContinuous_eq_one Λ).mpr h1]\n cases' det_eq_one_or_neg_one Λ' with h2 h2\n · rw [h2, (detContinuous_eq_one Λ').mpr h2]\n simp only [toMul_zero]\n · rw [h2, (detContinuous_eq_zero Λ').mpr h2]\n erw [Additive.toMul.apply_eq_iff_eq]\n change (0 : Fin 2) = (1 : Fin 2) ↔ _\n simp only [Fin.isValue, zero_ne_one, false_iff]\n linarith\n · rw [h1, (detContinuous_eq_zero Λ).mpr h1]\n cases' det_eq_one_or_neg_one Λ' with h2 h2\n · rw [h2, (detContinuous_eq_one Λ').mpr h2]\n erw [Additive.toMul.apply_eq_iff_eq]\n change (1 : Fin 2) = (0 : Fin 2) ↔ _\n simp only [Fin.isValue, one_ne_zero, false_iff]\n linarith\n · rw [h2, (detContinuous_eq_zero Λ').mpr h2]\n simp only [Nat.reduceAdd]",
"proofType": "tactic",
"proofLengthLines": 19,
"proofLengthTokens": 799
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import HepLean.SpaceTime.LorentzGroup.Basic
import Mathlib.RepresentationTheory.Basic
/-!
# Covariant Lorentz vectors
The type `LorentzVector` corresponds to contravariant Lorentz tensors.
In this section we define covariant Lorentz tensors.
-/
/-! TODO: Define equivariant map between contravariant and covariant lorentz tensors. -/
noncomputable section
open BigOperators
/- The number of space dimensions . -/
variable (d : ℕ)
/-- The type of covariant Lorentz Vectors in `d`-space dimensions. -/
def CovariantLorentzVector : Type := (Fin 1 ⊕ Fin d) → ℝ
/-- An instance of an additive commutative monoid on `LorentzVector`. -/
instance : AddCommMonoid (CovariantLorentzVector d) := Pi.addCommMonoid
/-- An instance of a module on `LorentzVector`. -/
noncomputable instance instModuleRealCovariantLorentzVector : Module ℝ (CovariantLorentzVector d) := Pi.module _ _ _
instance : AddCommGroup (CovariantLorentzVector d) := Pi.addCommGroup
/-- The structure of a topological space `LorentzVector d`. -/
instance : TopologicalSpace (CovariantLorentzVector d) :=
haveI : NormedAddCommGroup (CovariantLorentzVector d) := Pi.normedAddCommGroup
UniformSpace.toTopologicalSpace
namespace CovariantLorentzVector
variable {d : ℕ} (v : CovariantLorentzVector d)
/-- The standard basis of `LorentzVector` indexed by `Fin 1 ⊕ Fin (d)`. -/
@[simps!]
noncomputable def stdBasis : Basis (Fin 1 ⊕ Fin (d)) ℝ (CovariantLorentzVector d) := Pi.basisFun ℝ _
lemma decomp_stdBasis (v : CovariantLorentzVector d) : ∑ i, v i • stdBasis i = v := by
funext ν
rw [Finset.sum_apply, Finset.sum_eq_single_of_mem ν]
· simp only [HSMul.hSMul, SMul.smul, stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ ν
· intros b _ hbi
simp only [HSMul.hSMul, SMul.smul, stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [Pi.single]
apply Or.inr (Function.update_noteq (id (Ne.symm hbi)) 1 0)
| @[simp]
lemma decomp_stdBasis' (v : CovariantLorentzVector d) :
v (Sum.inl 0) • stdBasis (Sum.inl 0)
+ ∑ a₂ : Fin d, v (Sum.inr a₂) • stdBasis (Sum.inr a₂) = v | CovariantLorentzVector.decomp_stdBasis' | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Covariant.lean | HepLean.SpaceTime.LorentzVector.Covariant | HepLean.SpaceTime.LorentzVector.Covariant.jsonl | {
"lineInFile": 63,
"tokenPositionInFile": 2186,
"theoremPositionInFile": 3
} | {
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"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
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} | {
"hasProof": true,
"proof": ":= by\n trans ∑ i, v i • stdBasis i\n · simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,\n Finset.sum_singleton]\n · exact decomp_stdBasis v",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 180
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
| lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) | LorentzGroup.IsOrthochronous_iff_transpose | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 38,
"tokenPositionInFile": 1061,
"theoremPositionInFile": 2
} | {
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"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
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]
} | {
"hasProof": true,
"proof": ":= by rfl",
"proofType": "tactic",
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"proofLengthTokens": 9
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
simp only [unitUp, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
/-- The unit of the contraction of covariant Lorentz vector and a
contravariant Lorentz vector. -/
def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x := by
simp only [unitDown, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,
lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']
/-!
# Contractions and the Lorentz actions
-/
open Matrix
@[simp]
lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) := by
rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
| @[simp]
lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
(LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) | LorentzVector.contrDownUp_invariant_lorentzAction | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a65fb06605cbfe5b3907ae8a3963ce7cd664a975",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 160,
"tokenPositionInFile": 6033,
"theoremPositionInFile": 11
} | {
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"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
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"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
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"Mathlib.Algebra.Order.Monoid.Prod",
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"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
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"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
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"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
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"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
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"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
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"Mathlib.Data.Set.Intervals.Infinite",
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"Mathlib.GroupTheory.Finiteness",
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"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
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"Mathlib.Data.List.FinRange",
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"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
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"Mathlib.RingTheory.Ideal.Basic",
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"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
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"Mathlib.Data.Real.NNReal",
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"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
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"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
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"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
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"Mathlib.RingTheory.Adjoin.Basic",
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"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
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"Mathlib.Algebra.Algebra.Pi",
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"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
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"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
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"Mathlib.Data.W.Basic",
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"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
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"Mathlib.LinearAlgebra.Dimension.LinearMap",
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"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
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"Mathlib.Data.MvPolynomial.Rename",
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"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
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"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
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"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
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"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
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"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
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"Mathlib.Analysis.Convex.Normed",
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"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
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"Mathlib.Topology.MetricSpace.Completion",
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"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
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"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
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"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
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"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
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} | {
"hasProof": true,
"proof": ":= by\n rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]\n rw [dotProduct_comm, dotProduct_comm x y]\n simp only [rep_apply, CovariantLorentzVector.rep_apply]\n rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]\n simp only [LorentzGroup.subtype_inv_mul, one_mulVec]",
"proofType": "tactic",
"proofLengthLines": 5,
"proofLengthTokens": 297
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Proper
import Mathlib.Topology.Constructions
import HepLean.SpaceTime.LorentzVector.NormOne
/-!
# Boosts
This file defines those Lorentz which are boosts.
We first define generalised boosts, which are restricted lorentz transformations taking
a four velocity `u` to a four velocity `v`.
A boost is the special case of a generalised boost when `u = stdBasis 0`.
## References
- The main argument follows: Guillem Cobos, The Lorentz Group, 2015:
https://diposit.ub.edu/dspace/bitstream/2445/68763/2/memoria.pdf
-/
/-! TODO: Show that generalised boosts are in the restricted Lorentz group. -/
noncomputable section
namespace LorentzGroup
open NormOneLorentzVector
open minkowskiMetric
variable {d : ℕ}
/-- An auxillary linear map used in the definition of a generalised boost. -/
def genBoostAux₁ (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d where
toFun x := (2 * ⟪x, u⟫ₘ) • v.1.1
map_add' x y := by
simp only [map_add, LinearMap.add_apply]
rw [mul_add, add_smul]
map_smul' c x := by
simp only [map_smul, LinearMap.smul_apply, smul_eq_mul,
RingHom.id_apply]
rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
/-- An auxillary linear map used in the definition of a genearlised boost. -/
def genBoostAux₂ (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d where
toFun x := - (⟪x, u.1.1 + v⟫ₘ / (1 + ⟪u.1.1, v⟫ₘ)) • (u.1.1 + v)
map_add' x y := by
simp only
rw [← add_smul]
apply congrFun (congrArg _ _)
field_simp
apply congrFun (congrArg _ _)
ring
map_smul' c x := by
simp only
rw [map_smul, show (c • minkowskiMetric x) (↑u + ↑v) = c * minkowskiMetric x (u+v) from rfl,
mul_div_assoc, neg_mul_eq_mul_neg, smul_smul]
rfl
/-- An generalised boost. This is a Lorentz transformation which takes the four velocity `u`
to `v`. -/
def genBoost (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d :=
LinearMap.id + genBoostAux₁ u v + genBoostAux₂ u v
namespace genBoost
| /--
This lemma states that for a given four-velocity `u`, the general boost
transformation `genBoost u u` is equal to the identity linear map `LinearMap.id`.
-/
lemma self (u : FuturePointing d) : genBoost u u = LinearMap.id | LorentzGroup.genBoost.self | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "28cc0ff7121fba90f032469b1685b796e604635b",
"date": "2024-04-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Boosts.lean | HepLean.SpaceTime.LorentzGroup.Boosts | HepLean.SpaceTime.LorentzGroup.Boosts.jsonl | {
"lineInFile": 69,
"tokenPositionInFile": 2202,
"theoremPositionInFile": 1
} | {
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"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
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"Mathlib.GroupTheory.Subsemigroup.Membership",
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"Mathlib.Order.Filter.Extr",
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"Mathlib.Algebra.Module.Submodule.LinearMap",
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"Mathlib.Tactic.IntervalCases",
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"Mathlib.Order.SupClosed",
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"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
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"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
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"Mathlib.Algebra.Module.Submodule.Bilinear",
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"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
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"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.LinearAlgebra.Pi",
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"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
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"Mathlib.Algebra.Order.Field.InjSurj",
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"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
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"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
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"Mathlib.LinearAlgebra.AffineSpace.Basic",
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"Mathlib.GroupTheory.Submonoid.Order",
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"Mathlib.LinearAlgebra.Matrix.Trace",
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"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
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"Mathlib.RingTheory.Congruence",
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"Mathlib.RingTheory.Finiteness",
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"Mathlib.RingTheory.Nilpotent",
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"Mathlib.RingTheory.UniqueFactorizationDomain",
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"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
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"Mathlib.Algebra.GCDMonoid.Multiset",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
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"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
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"Mathlib.Data.Finset.Fin",
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"Mathlib.GroupTheory.Perm.Finite",
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"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
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"Mathlib.LinearAlgebra.Multilinear.Basic",
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"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
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"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
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"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
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"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
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"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
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"Mathlib.RingTheory.NonUnitalSubring.Basic",
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"Mathlib.Topology.MetricSpace.Dilation",
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"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
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"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
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"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
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"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
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"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
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"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
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"Mathlib.Topology.GDelta",
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"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
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} | {
"hasProof": true,
"proof": ":= by\n ext x\n simp only [genBoost, LinearMap.add_apply, LinearMap.id_coe, id_eq]\n rw [add_assoc, add_right_eq_self, add_eq_zero_iff_eq_neg, genBoostAux₁, genBoostAux₂]\n simp only [LinearMap.coe_mk, AddHom.coe_mk, map_add, smul_add, neg_smul, neg_add_rev, neg_neg]\n rw [← add_smul]\n apply congrFun (congrArg _ _)\n rw [u.1.2]\n ring",
"proofType": "tactic",
"proofLengthLines": 8,
"proofLengthTokens": 337
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
lemma IsOrthochronous_iff_ge_one :
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
NormOneLorentzVector.time_pos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_neg_one :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ -1 := by
rw [timeComp, IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.not_mem_iff,
NormOneLorentzVector.time_nonpos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_zero :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [not_orthochronous_iff_le_neg_one] at h
linarith
· rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
def stepFunction : ℝ → ℝ := fun t =>
if t ≤ -1 then -1 else
if 1 ≤ t then 1 else t
lemma stepFunction_continuous : Continuous stepFunction := by
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
<;> intro a ha
· rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp only [le_neg_self_iff, id_eq]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
· rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
exact id (Eq.symm ha)
/-- The continuous map from `lorentzGroup` to `ℝ` wh
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
def orthchroMapReal : C(LorentzGroup d, ℝ) := ContinuousMap.comp
⟨stepFunction, stepFunction_continuous⟩ timeCompCont
lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMapReal Λ = 1 := by
rw [IsOrthochronous_iff_ge_one, timeComp] at h
change stepFunction (Λ.1 _ _) = 1
rw [stepFunction, if_pos h, if_neg]
linarith
lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMapReal Λ = - 1 := by
rw [not_orthochronous_iff_le_neg_one] at h
change stepFunction (timeComp _)= - 1
rw [stepFunction, if_pos h]
lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
by_cases h : IsOrthochronous Λ
· exact Or.inr $ orthchroMapReal_on_IsOrthochronous h
· exact Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
local notation "ℤ₂" => Multiplicative (ZMod 2)
/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
def orthchroMap : C(LorentzGroup d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMap Λ = 1 := by
simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
| lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMap Λ = Additive.toMul (1 : ZMod 2) | LorentzGroup.orthchroMap_not_IsOrthochronous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 121,
"tokenPositionInFile": 4526,
"theoremPositionInFile": 15
} | {
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"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
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"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
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"Mathlib.Data.Set.Intervals.Infinite",
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"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
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"Mathlib.Data.Finsupp.Fin",
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"Mathlib.Algebra.Algebra.Hom",
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"Mathlib.Algebra.Module.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
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"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
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"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.LinearAlgebra.Pi",
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"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
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"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
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"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
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"Mathlib.Algebra.Ring.OrderSynonym",
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"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
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"Mathlib.SetTheory.Ordinal.Exponential",
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"Mathlib.SetTheory.Cardinal.Ordinal",
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"Mathlib.LinearAlgebra.AffineSpace.Basic",
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"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
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"Mathlib.Algebra.DirectSum.Module",
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"Mathlib.Topology.EMetricSpace.Lipschitz",
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"Mathlib.SetTheory.Cardinal.Continuum",
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"Mathlib.RingTheory.AlgebraTower",
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"Mathlib.Data.Matrix.Basic",
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"Mathlib.LinearAlgebra.Matrix.Trace",
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"Mathlib.LinearAlgebra.StdBasis",
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"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
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"Mathlib.GroupTheory.NoncommPiCoprod",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
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"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
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"Mathlib.Algebra.FreeAlgebra",
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"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
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"Mathlib.GroupTheory.Subgroup.Simple",
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"Mathlib.GroupTheory.PGroup",
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"Mathlib.LinearAlgebra.Dimension.DivisionRing",
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"Mathlib.LinearAlgebra.Dimension.LinearMap",
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"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
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"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
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"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
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"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
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"Mathlib.Data.Polynomial.Degree.Lemmas",
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"Mathlib.Data.Polynomial.Reverse",
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"Mathlib.RingTheory.EuclideanDomain",
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"Mathlib.Tactic.ComputeDegree",
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"Mathlib.Logic.Equiv.TransferInstance",
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"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
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"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
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"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
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"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
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"Mathlib.Data.Matrix.DMatrix",
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"Mathlib.Topology.MetricSpace.Completion",
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"Mathlib.Analysis.Seminorm",
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"Mathlib.RingTheory.Artinian",
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"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
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"Mathlib.Tactic.NoncommRing",
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"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"HepLean.SpaceTime.LorentzGroup.Proper"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [orthchroMap, ContinuousMap.comp_apply, ContinuousMap.coe_mk,\n orthchroMapReal_on_not_IsOrthochronous h, coeForℤ₂_apply, Subtype.mk.injEq, Nat.reduceAdd]\n rw [if_neg]\n · rfl\n · linarith",
"proofType": "tactic",
"proofLengthLines": 5,
"proofLengthTokens": 209
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
@[simp]
lemma dual_id : @dual d 1 = 1 := by
simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
| @[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ | minkowskiMetric.dual_mul | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
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} | {
"hasProof": true,
"proof": ":= by\n simp only [dual, transpose_mul]\n trans η * Λ'ᵀ * (η * η) * Λᵀ * η\n · noncomm_ring [minkowskiMatrix.sq]\n · noncomm_ring",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 129
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.Algebra.Lie.Classical
/-!
# The Lorentz Algebra
We define
- Define `lorentzAlgebra` via `LieAlgebra.Orthogonal.so'` as a subalgebra of
`Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`.
- In `mem_iff` prove that a matrix is in the Lorentz algebra if and only if it satisfies the
condition `Aᵀ * η = - η * A`.
-/
namespace SpaceTime
open Matrix
open TensorProduct
/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`. -/
def lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :=
(LieAlgebra.Orthogonal.so' (Fin 1) (Fin 3) ℝ)
namespace lorentzAlgebra
open minkowskiMatrix
lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
have h1 := A.2
erw [mem_skewAdjointMatricesLieSubalgebra] at h1
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h1
lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ}
(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra := by
erw [mem_skewAdjointMatricesLieSubalgebra]
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h
| lemma mem_iff {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} :
A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A | SpaceTime.lorentzAlgebra.mem_iff | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | {
"commit": "8fd0b63edbfd2814d1fba639a9838632881f48e4",
"date": "2024-05-09T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzAlgebra/Basic.lean | HepLean.SpaceTime.LorentzAlgebra.Basic | HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl | {
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"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
| @[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 | minkowskiMatrix.sq | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 37,
"tokenPositionInFile": 907,
"theoremPositionInFile": 2
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"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
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"hasProof": true,
"proof": ":= by\n simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]\n ext1 i j\n rcases i with i | i <;> rcases j with j | j\n · simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]\n split\n · rename_i h\n subst h\n simp_all only [one_apply_eq]\n · simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]\n · rfl\n · rfl\n · simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]\n split\n · rename_i h\n subst h\n simp_all only [one_apply_eq]\n · simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]",
"proofType": "tactic",
"proofLengthLines": 17,
"proofLengthTokens": 641
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
lemma IsOrthochronous_iff_ge_one :
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
NormOneLorentzVector.time_pos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_neg_one :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ -1 := by
rw [timeComp, IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.not_mem_iff,
NormOneLorentzVector.time_nonpos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_zero :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [not_orthochronous_iff_le_neg_one] at h
linarith
· rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
def stepFunction : ℝ → ℝ := fun t =>
if t ≤ -1 then -1 else
if 1 ≤ t then 1 else t
lemma stepFunction_continuous : Continuous stepFunction := by
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
<;> intro a ha
· rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp only [le_neg_self_iff, id_eq]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
· rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
exact id (Eq.symm ha)
/-- The continuous map from `lorentzGroup` to `ℝ` wh
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
def orthchroMapReal : C(LorentzGroup d, ℝ) := ContinuousMap.comp
⟨stepFunction, stepFunction_continuous⟩ timeCompCont
| lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMapReal Λ = 1 | LorentzGroup.orthchroMapReal_on_IsOrthochronous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"theoremPositionInFile": 10
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"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
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"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
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"Mathlib.Order.Filter.Lift",
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"Mathlib.Data.Set.BoolIndicator",
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"Mathlib.Data.Fintype.BigOperators",
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"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
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"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
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"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
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"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
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"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
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"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
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"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
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"Mathlib.Data.Nat.Prime",
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"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
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"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
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"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
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"Mathlib.Order.Filter.Extr",
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"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
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"Mathlib.Data.List.Palindrome",
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"Mathlib.Data.Nat.Digits",
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"Mathlib.Data.Set.Intervals.Infinite",
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"Mathlib.GroupTheory.Finiteness",
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"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
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"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
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"Mathlib.Algebra.Module.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
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"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
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"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
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"Mathlib.RingTheory.Ideal.Basic",
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"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
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"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
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"Mathlib.Algebra.Ring.OrderSynonym",
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"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
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"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
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"Mathlib.SetTheory.Cardinal.Ordinal",
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"Mathlib.Topology.EMetricSpace.Basic",
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"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
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"Mathlib.Topology.Instances.Int",
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"Mathlib.Data.Complex.Abs",
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"Mathlib.LinearAlgebra.AffineSpace.Basic",
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"Mathlib.GroupTheory.Submonoid.Order",
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"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
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"Mathlib.Analysis.SpecificLimits.Basic",
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"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
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"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
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"Mathlib.Algebra.Algebra.Pi",
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"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
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"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
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"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
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"Mathlib.GroupTheory.NoncommPiCoprod",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.Factorization.Basic",
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"Mathlib.GroupTheory.Exponent",
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"Mathlib.Data.Fintype.Perm",
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"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"HepLean.SpaceTime.LorentzGroup.Proper"
]
} | {
"hasProof": true,
"proof": ":= by\n rw [IsOrthochronous_iff_ge_one, timeComp] at h\n change stepFunction (Λ.1 _ _) = 1\n rw [stepFunction, if_pos h, if_neg]\n linarith",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 139
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
simp only [unitUp, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
/-- The unit of the contraction of covariant Lorentz vector and a
contravariant Lorentz vector. -/
def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x := by
simp only [unitDown, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,
lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']
/-!
# Contractions and the Lorentz actions
-/
open Matrix
@[simp]
lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) := by
rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
@[simp]
lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
(LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) := by
rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]
rw [dotProduct_comm, dotProduct_comm x y]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
end LorentzVector
namespace minkowskiMatrix
open LorentzVector
open TensorStructure
open scoped minkowskiMatrix
variable {d : ℕ}
/-- The metric tensor as an element of `LorentzVector d ⊗[ℝ] LorentzVector d`. -/
def asTenProd : LorentzVector d ⊗[ℝ] LorentzVector d :=
∑ μ, ∑ ν, η μ ν • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis ν)
lemma asTenProd_diag :
@asTenProd d = ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ) := by
simp only [asTenProd]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [Finset.sum_eq_single μ]
· intro ν _ hμν
rw [minkowskiMatrix.off_diag_zero hμν.symm]
exact TensorProduct.zero_smul (e μ ⊗ₜ[ℝ] e ν)
· intro a
rename_i j
exact False.elim (a j)
/-- The metric tensor as an element of `CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d`. -/
def asCoTenProd : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ μ, ∑ ν, η μ ν • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν)
lemma asCoTenProd_diag :
@asCoTenProd d = ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ]
CovariantLorentzVector.stdBasis μ) := by
simp only [asCoTenProd]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [Finset.sum_eq_single μ]
· intro ν _ hμν
rw [minkowskiMatrix.off_diag_zero hμν.symm]
simp only [zero_smul]
· intro a
simp_all only
@[simp]
lemma contrLeft_asTenProd (x : CovariantLorentzVector d) :
contrLeftAux contrDownUp (x ⊗ₜ[ℝ] asTenProd) =
∑ μ, ((η μ μ * x μ) • LorentzVector.stdBasis μ) := by
simp only [asTenProd_diag]
rw [tmul_sum]
rw [map_sum]
refine Finset.sum_congr rfl (fun μ _ => ?_)
simp only [contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq, lid_tmul]
exact smul_smul (η μ μ) (x μ) (e μ)
| @[simp]
lemma contrLeft_asCoTenProd (x : LorentzVector d) :
contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asCoTenProd) =
∑ μ, ((η μ μ * x μ) • CovariantLorentzVector.stdBasis μ) | minkowskiMatrix.contrLeft_asCoTenProd | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 222,
"tokenPositionInFile": 8460,
"theoremPositionInFile": 17
} | {
"inFilePremises": true,
"numInFilePremises": 4,
"repositoryPremises": true,
"numRepositoryPremises": 15,
"numPremises": 131,
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"Init.MetaTypes",
"Init.SimpLemmas",
"Init.Data.Nat.Basic",
"Init.WF",
"Init.WFTactics",
"Init.Data.Nat.Div",
"Init.Data.Nat.Bitwise.Basic",
"Init.Data.Fin.Basic",
"Init.Data.UInt.Basic",
"Init.Control.Basic",
"Init.Control.Id",
"Init.Control.Except",
"Init.Control.State",
"Init.Data.Cast",
"Init.Data.List.Basic",
"Init.Data.Int.Basic",
"Init.Data.Char.Basic",
"Init.Data.Option.Basic",
"Init.Data.String.Basic",
"Init.Data.Format.Basic",
"Init.Data.Repr",
"Init.Control.Option",
"Init.Data.ToString.Basic",
"Init.Util",
"Init.Data.Array.Basic",
"Init.Data.Option.BasicAux",
"Init.Data.Array.Subarray",
"Init.Data.ByteArray.Basic",
"Init.Data.ByteArray",
"Init.Data.String.Extra",
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"Init.Data.ToString",
"Init.NotationExtra",
"Init.TacticsExtra",
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"Init.RCases",
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"Init.Data.Int.DivModLemmas",
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"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
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"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
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"Mathlib.Logic.IsEmpty",
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"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
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"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
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"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
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"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
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"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
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"Aesop.BuiltinRules.Split",
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"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
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"Aesop.Main",
"Aesop",
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"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
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"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
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"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
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"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
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"Qq.SortLocalDecls",
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"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
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"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
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"Mathlib.Tactic.ApplyWith",
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"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
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"Mathlib.Tactic.SimpIntro",
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"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
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"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
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"Mathlib.Logic.Equiv.Defs",
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"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
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"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
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"Mathlib.Algebra.Order.Group.OrderIso",
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"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
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"Mathlib.Logic.Equiv.Set",
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"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
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"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
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"Mathlib.Algebra.Group.Pi.Lemmas",
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"Mathlib.Data.Multiset.Range",
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"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
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"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
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"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
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"Mathlib.Data.List.OfFn",
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"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
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"Mathlib.Data.Set.Sigma",
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"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic",
"HepLean.SpaceTime.LorentzVector.LorentzAction",
"HepLean.SpaceTime.LorentzVector.Covariant",
"Mathlib.LinearAlgebra.Multilinear.TensorProduct",
"Mathlib.LinearAlgebra.PiTensorProduct",
"HepLean.Tensors.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [asCoTenProd_diag]\n rw [tmul_sum]\n rw [map_sum]\n refine Finset.sum_congr rfl (fun μ _ => ?_)\n simp only [contrLeftAux, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,\n LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,\n contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul]\n exact smul_smul (η μ μ) (x μ) (CovariantLorentzVector.stdBasis μ)",
"proofType": "tactic",
"proofLengthLines": 8,
"proofLengthTokens": 418
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
| lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 | minkowskiMetric.time_sq_eq_metric_add_space | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 163,
"tokenPositionInFile": 5079,
"theoremPositionInFile": 14
} | {
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"Mathlib.Tactic.SudoSetOption",
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"ProofWidgets.Component.Basic",
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"ProofWidgets.Data.Html",
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"ProofWidgets.Component.OfRpcMethod",
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"Mathlib.Util.CountHeartbeats",
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"Mathlib.Order.Lattice",
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"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
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]
} | {
"hasProof": true,
"proof": ":= by\n rw [self_eq_time_minus_norm]\n exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 98
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import HepLean.SpaceTime.LorentzGroup.Basic
import Mathlib.RepresentationTheory.Basic
/-!
# Lorentz group action on Lorentz vectors.
-/
noncomputable section
open BigOperators
namespace LorentzVector
variable {d : ℕ} (v : LorentzVector d)
/-- The contravariant action of the Lorentz group on a Lorentz vector. -/
def rep : Representation ℝ (LorentzGroup d) (LorentzVector d) where
toFun g := Matrix.toLinAlgEquiv e g
map_one' := (MulEquivClass.map_eq_one_iff (Matrix.toLinAlgEquiv e)).mpr rfl
map_mul' x y := by
simp only [lorentzGroupIsGroup_mul_coe, map_mul]
open Matrix in
| lemma rep_apply (g : LorentzGroup d) : rep g v = g *ᵥ v | LorentzVector.rep_apply | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a65fb06605cbfe5b3907ae8a3963ce7cd664a975",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/LorentzAction.lean | HepLean.SpaceTime.LorentzVector.LorentzAction | HepLean.SpaceTime.LorentzVector.LorentzAction.jsonl | {
"lineInFile": 30,
"tokenPositionInFile": 800,
"theoremPositionInFile": 0
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"Lean.Elab.Tactic.Omega.Frontend",
"Lean.Elab.Tactic.Omega",
"Lean.Elab.Tactic.Simpa",
"Lean.Meta.Tactic.NormCast",
"Lean.Elab.Tactic.NormCast",
"Lean.Elab.Tactic.Symm",
"Lean.LabelAttribute",
"Lean.Meta.Iterator",
"Lean.Meta.Tactic.IndependentOf",
"Lean.Meta.Tactic.Backtrack",
"Lean.Meta.Tactic.SolveByElim",
"Lean.Elab.Tactic.SolveByElim",
"Lean.Meta.LazyDiscrTree",
"Lean.Util.Heartbeats",
"Lean.Meta.Tactic.LibrarySearch",
"Lean.Elab.Tactic.LibrarySearch",
"Lean.Elab.Tactic.ShowTerm",
"Lean.Elab.Tactic",
"Lean.Elab.StructInst",
"Lean.Elab.Print",
"Lean.Elab.PreDefinition.WF",
"Lean.Elab.PreDefinition",
"Lean.Elab.Deriving.Util",
"Lean.Elab.Deriving.Inhabited",
"Lean.Elab.Deriving.Nonempty",
"Lean.Elab.Deriving.TypeName",
"Lean.Elab.Deriving.BEq",
"Lean.Meta.Inductive",
"Lean.Elab.Deriving.DecEq",
"Lean.Elab.Deriving.Repr",
"Lean.Elab.Deriving.FromToJson",
"Lean.Elab.Deriving.SizeOf",
"Lean.Elab.Deriving.Hashable",
"Lean.Elab.Deriving.Ord",
"Lean.Elab.Deriving",
"Lean.Elab.Extra",
"Lean.Elab.GenInjective",
"Lean.Elab.Macro",
"Lean.Elab.Notation",
"Lean.Elab.Mixfix",
"Lean.Elab.MacroRules",
"Lean.Elab.BuiltinCommand",
"Lean.Elab.InheritDoc",
"Lean.Elab.ParseImportsFast",
"Lean.Elab.GuardMsgs",
"Lean.Elab.CheckTactic",
"Lean.Elab.MatchExpr",
"Lean.Elab",
"Lean.Meta.LevelDefEq",
"Lean.Meta.UnificationHint",
"Lean.Meta.ExprDefEq",
"Lean.Meta.Tactic.LinearArith.Solver",
"Lean.Meta.Tactic.LinearArith.Nat.Solver",
"Lean.Meta.Tactic.LinearArith.Nat",
"Lean.Meta.Tactic.LinearArith.Main",
"Lean.Meta.Tactic.LinearArith",
"Lean.Meta.Tactic.AC.Main",
"Lean.Meta.Tactic.AC",
"Lean.Meta.Tactic",
"Lean.Meta.ExprLens",
"Lean.Meta.ExprTraverse",
"Lean.Meta",
"Lean.Util.ShareCommon",
"Lean.Util.TestExtern",
"Lean.Util.LeanOptions",
"Lean.Util.FileSetupInfo",
"Lean.Util",
"Lean.Server.Watchdog",
"Lean.LoadDynlib",
"Lean.Server.FileWorker.WidgetRequests",
"Lean.Util.LakePath",
"Lean.Server.FileWorker.SetupFile",
"Lean.Server.ImportCompletion",
"Lean.Server.FileWorker",
"Lean.Server.Rpc.Deriving",
"Lean.Server.Rpc",
"Lean.Server",
"Lean.Widget",
"Lean.Linter.Builtin",
"Lean.Linter",
"Lean",
"Std.Classes.BEq",
"Std.Classes.Cast",
"Std.Classes.Order",
"Std.Classes.RatCast",
"Std.Classes.SatisfiesM",
"Std.CodeAction.Misc",
"Std.CodeAction",
"Std.Control.ForInStep.Basic",
"Std.Control.ForInStep.Lemmas",
"Std.Control.ForInStep",
"Std.Control.Lemmas",
"Std.Data.MLList.Basic",
"Std.Control.Nondet.Basic",
"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
"Std.Data.Bool",
"Std.Data.Fin.Basic",
"Std.Data.Option.Lemmas",
"Std.Data.List.Lemmas",
"Std.Tactic.SeqFocus",
"Std.Util.ProofWanted",
"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
"Std.Data.BinomialHeap.Basic",
"Std.Data.BinomialHeap.Lemmas",
"Std.Data.BinomialHeap",
"Std.Data.Fin.Lemmas",
"Std.Data.BitVec.Lemmas",
"Std.Data.BitVec",
"Std.Data.ByteArray",
"Std.Data.Char",
"Std.Data.DList",
"Std.Data.Fin",
"Std.Data.HashMap.Basic",
"Std.Data.HashMap.Lemmas",
"Std.Data.HashMap.WF",
"Std.Data.HashMap",
"Std.Data.Int.Gcd",
"Std.Data.Int.Lemmas",
"Std.Data.Int",
"Std.Data.LazyList",
"Std.Data.List.Count",
"Std.Data.List.Pairwise",
"Std.Data.List.Perm",
"Std.Data.List",
"Std.Data.MLList.Heartbeats",
"Std.Lean.System.IO",
"Std.Data.MLList.IO",
"Std.Data.MLList",
"Std.Data.Nat",
"Std.Data.Option",
"Std.Data.PairingHeap",
"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
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"Mathlib.RingTheory.Ideal.Quotient",
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"Mathlib.LinearAlgebra.Matrix.ToLin",
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"Mathlib.Algebra.Group.UniqueProds",
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"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
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"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
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"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
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"Mathlib.LinearAlgebra.Matrix.Polynomial",
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"Mathlib.Data.Matrix.DMatrix",
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"Mathlib.LinearAlgebra.Matrix.Reindex",
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"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
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"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
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"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
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"Mathlib.Analysis.NormedSpace.AddTorsor",
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"Mathlib.Topology.Algebra.FilterBasis",
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"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
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"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
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"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
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"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
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"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic"
]
} | {
"hasProof": true,
"proof": ":= rfl",
"proofType": "term",
"proofLengthLines": 0,
"proofLengthTokens": 6
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Proper
import Mathlib.Topology.Constructions
import HepLean.SpaceTime.LorentzVector.NormOne
/-!
# Boosts
This file defines those Lorentz which are boosts.
We first define generalised boosts, which are restricted lorentz transformations taking
a four velocity `u` to a four velocity `v`.
A boost is the special case of a generalised boost when `u = stdBasis 0`.
## References
- The main argument follows: Guillem Cobos, The Lorentz Group, 2015:
https://diposit.ub.edu/dspace/bitstream/2445/68763/2/memoria.pdf
-/
/-! TODO: Show that generalised boosts are in the restricted Lorentz group. -/
noncomputable section
namespace LorentzGroup
open NormOneLorentzVector
open minkowskiMetric
variable {d : ℕ}
/-- An auxillary linear map used in the definition of a generalised boost. -/
def genBoostAux₁ (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d where
toFun x := (2 * ⟪x, u⟫ₘ) • v.1.1
map_add' x y := by
simp only [map_add, LinearMap.add_apply]
rw [mul_add, add_smul]
map_smul' c x := by
simp only [map_smul, LinearMap.smul_apply, smul_eq_mul,
RingHom.id_apply]
rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
/-- An auxillary linear map used in the definition of a genearlised boost. -/
def genBoostAux₂ (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d where
toFun x := - (⟪x, u.1.1 + v⟫ₘ / (1 + ⟪u.1.1, v⟫ₘ)) • (u.1.1 + v)
map_add' x y := by
simp only
rw [← add_smul]
apply congrFun (congrArg _ _)
field_simp
apply congrFun (congrArg _ _)
ring
map_smul' c x := by
simp only
rw [map_smul, show (c • minkowskiMetric x) (↑u + ↑v) = c * minkowskiMetric x (u+v) from rfl,
mul_div_assoc, neg_mul_eq_mul_neg, smul_smul]
rfl
/-- An generalised boost. This is a Lorentz transformation which takes the four velocity `u`
to `v`. -/
def genBoost (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d :=
LinearMap.id + genBoostAux₁ u v + genBoostAux₂ u v
namespace genBoost
/--
This lemma states that for a given four-velocity `u`, the general boost
transformation `genBoost u u` is equal to the identity linear map `LinearMap.id`.
-/
lemma self (u : FuturePointing d) : genBoost u u = LinearMap.id := by
ext x
simp only [genBoost, LinearMap.add_apply, LinearMap.id_coe, id_eq]
rw [add_assoc, add_right_eq_self, add_eq_zero_iff_eq_neg, genBoostAux₁, genBoostAux₂]
simp only [LinearMap.coe_mk, AddHom.coe_mk, map_add, smul_add, neg_smul, neg_add_rev, neg_neg]
rw [← add_smul]
apply congrFun (congrArg _ _)
rw [u.1.2]
ring
/-- A generalised boost as a matrix. -/
def toMatrix (u v : FuturePointing d) : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LinearMap.toMatrix LorentzVector.stdBasis LorentzVector.stdBasis (genBoost u v)
| lemma toMatrix_mulVec (u v : FuturePointing d) (x : LorentzVector d) :
(toMatrix u v).mulVec x = genBoost u v x | LorentzGroup.genBoost.toMatrix_mulVec | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "fdf5fda1e73e043ba40a3836ed1f00ddcc259373",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Boosts.lean | HepLean.SpaceTime.LorentzGroup.Boosts | HepLean.SpaceTime.LorentzGroup.Boosts.jsonl | {
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"theoremPositionInFile": 3
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"Lean.Elab.Tactic.Conv.Delta",
"Lean.Meta.Tactic.Unfold",
"Lean.Elab.Tactic.Unfold",
"Lean.Elab.Tactic.Conv.Unfold",
"Lean.Elab.Tactic.Conv",
"Lean.Elab.Tactic.Meta",
"Lean.Elab.Tactic.Cache",
"Lean.Elab.Calc",
"Lean.Elab.Tactic.Calc",
"Lean.Elab.Tactic.Congr",
"Lean.Elab.Tactic.Guard",
"Lean.Elab.Tactic.Change",
"Lean.Elab.Tactic.FalseOrByContra",
"Lean.Elab.Tactic.Omega.OmegaM",
"Lean.Elab.Tactic.Omega.MinNatAbs",
"Lean.Elab.Tactic.Omega.Core",
"Lean.Elab.Tactic.Omega.Frontend",
"Lean.Elab.Tactic.Omega",
"Lean.Elab.Tactic.Simpa",
"Lean.Meta.Tactic.NormCast",
"Lean.Elab.Tactic.NormCast",
"Lean.Elab.Tactic.Symm",
"Lean.LabelAttribute",
"Lean.Meta.Iterator",
"Lean.Meta.Tactic.IndependentOf",
"Lean.Meta.Tactic.Backtrack",
"Lean.Meta.Tactic.SolveByElim",
"Lean.Elab.Tactic.SolveByElim",
"Lean.Meta.LazyDiscrTree",
"Lean.Util.Heartbeats",
"Lean.Meta.Tactic.LibrarySearch",
"Lean.Elab.Tactic.LibrarySearch",
"Lean.Elab.Tactic.ShowTerm",
"Lean.Elab.Tactic",
"Lean.Elab.StructInst",
"Lean.Elab.Print",
"Lean.Elab.PreDefinition.WF",
"Lean.Elab.PreDefinition",
"Lean.Elab.Deriving.Util",
"Lean.Elab.Deriving.Inhabited",
"Lean.Elab.Deriving.Nonempty",
"Lean.Elab.Deriving.TypeName",
"Lean.Elab.Deriving.BEq",
"Lean.Meta.Inductive",
"Lean.Elab.Deriving.DecEq",
"Lean.Elab.Deriving.Repr",
"Lean.Elab.Deriving.FromToJson",
"Lean.Elab.Deriving.SizeOf",
"Lean.Elab.Deriving.Hashable",
"Lean.Elab.Deriving.Ord",
"Lean.Elab.Deriving",
"Lean.Elab.Extra",
"Lean.Elab.GenInjective",
"Lean.Elab.Macro",
"Lean.Elab.Notation",
"Lean.Elab.Mixfix",
"Lean.Elab.MacroRules",
"Lean.Elab.BuiltinCommand",
"Lean.Elab.InheritDoc",
"Lean.Elab.ParseImportsFast",
"Lean.Elab.GuardMsgs",
"Lean.Elab.CheckTactic",
"Lean.Elab.MatchExpr",
"Lean.Elab",
"Lean.Meta.LevelDefEq",
"Lean.Meta.UnificationHint",
"Lean.Meta.ExprDefEq",
"Lean.Meta.Tactic.LinearArith.Solver",
"Lean.Meta.Tactic.LinearArith.Nat.Solver",
"Lean.Meta.Tactic.LinearArith.Nat",
"Lean.Meta.Tactic.LinearArith.Main",
"Lean.Meta.Tactic.LinearArith",
"Lean.Meta.Tactic.AC.Main",
"Lean.Meta.Tactic.AC",
"Lean.Meta.Tactic",
"Lean.Meta.ExprLens",
"Lean.Meta.ExprTraverse",
"Lean.Meta",
"Lean.Util.ShareCommon",
"Lean.Util.TestExtern",
"Lean.Util.LeanOptions",
"Lean.Util.FileSetupInfo",
"Lean.Util",
"Lean.Server.Watchdog",
"Lean.LoadDynlib",
"Lean.Server.FileWorker.WidgetRequests",
"Lean.Util.LakePath",
"Lean.Server.FileWorker.SetupFile",
"Lean.Server.ImportCompletion",
"Lean.Server.FileWorker",
"Lean.Server.Rpc.Deriving",
"Lean.Server.Rpc",
"Lean.Server",
"Lean.Widget",
"Lean.Linter.Builtin",
"Lean.Linter",
"Lean",
"Std.Classes.BEq",
"Std.Classes.Cast",
"Std.Classes.Order",
"Std.Classes.RatCast",
"Std.Classes.SatisfiesM",
"Std.CodeAction.Misc",
"Std.CodeAction",
"Std.Control.ForInStep.Basic",
"Std.Control.ForInStep.Lemmas",
"Std.Control.ForInStep",
"Std.Control.Lemmas",
"Std.Data.MLList.Basic",
"Std.Control.Nondet.Basic",
"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
"Std.Data.Bool",
"Std.Data.Fin.Basic",
"Std.Data.Option.Lemmas",
"Std.Data.List.Lemmas",
"Std.Tactic.SeqFocus",
"Std.Util.ProofWanted",
"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
"Std.Data.BinomialHeap.Basic",
"Std.Data.BinomialHeap.Lemmas",
"Std.Data.BinomialHeap",
"Std.Data.Fin.Lemmas",
"Std.Data.BitVec.Lemmas",
"Std.Data.BitVec",
"Std.Data.ByteArray",
"Std.Data.Char",
"Std.Data.DList",
"Std.Data.Fin",
"Std.Data.HashMap.Basic",
"Std.Data.HashMap.Lemmas",
"Std.Data.HashMap.WF",
"Std.Data.HashMap",
"Std.Data.Int.Gcd",
"Std.Data.Int.Lemmas",
"Std.Data.Int",
"Std.Data.LazyList",
"Std.Data.List.Count",
"Std.Data.List.Pairwise",
"Std.Data.List.Perm",
"Std.Data.List",
"Std.Data.MLList.Heartbeats",
"Std.Lean.System.IO",
"Std.Data.MLList.IO",
"Std.Data.MLList",
"Std.Data.Nat",
"Std.Data.Option",
"Std.Data.PairingHeap",
"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
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"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
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"Mathlib.Topology.MetricSpace.DilationEquiv",
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"HepLean.SpaceTime.MinkowskiMetric",
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"HepLean.SpaceTime.LorentzGroup.Basic",
"HepLean.SpaceTime.LorentzGroup.Proper"
]
} | {
"hasProof": true,
"proof": ":=\n LinearMap.toMatrix_mulVec_repr LorentzVector.stdBasis LorentzVector.stdBasis (genBoost u v) x",
"proofType": "term",
"proofLengthLines": 1,
"proofLengthTokens": 98
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
lemma IsOrthochronous_iff_ge_one :
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
NormOneLorentzVector.time_pos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_neg_one :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ -1 := by
rw [timeComp, IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.not_mem_iff,
NormOneLorentzVector.time_nonpos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_zero :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [not_orthochronous_iff_le_neg_one] at h
linarith
· rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
def stepFunction : ℝ → ℝ := fun t =>
if t ≤ -1 then -1 else
if 1 ≤ t then 1 else t
lemma stepFunction_continuous : Continuous stepFunction := by
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
<;> intro a ha
· rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp only [le_neg_self_iff, id_eq]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
· rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
exact id (Eq.symm ha)
/-- The continuous map from `lorentzGroup` to `ℝ` wh
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
def orthchroMapReal : C(LorentzGroup d, ℝ) := ContinuousMap.comp
⟨stepFunction, stepFunction_continuous⟩ timeCompCont
lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMapReal Λ = 1 := by
rw [IsOrthochronous_iff_ge_one, timeComp] at h
change stepFunction (Λ.1 _ _) = 1
rw [stepFunction, if_pos h, if_neg]
linarith
lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMapReal Λ = - 1 := by
rw [not_orthochronous_iff_le_neg_one] at h
change stepFunction (timeComp _)= - 1
rw [stepFunction, if_pos h]
lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
by_cases h : IsOrthochronous Λ
· exact Or.inr $ orthchroMapReal_on_IsOrthochronous h
· exact Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
local notation "ℤ₂" => Multiplicative (ZMod 2)
/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
def orthchroMap : C(LorentzGroup d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMap Λ = 1 := by
simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
simp only [orthchroMap, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
orthchroMapReal_on_not_IsOrthochronous h, coeForℤ₂_apply, Subtype.mk.injEq, Nat.reduceAdd]
rw [if_neg]
· rfl
· linarith
lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
(h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
rw [IsOrthochronous_iff_transpose] at h
rw [IsOrthochronous_iff_futurePointing] at h h'
rw [IsOrthochronous, timeComp_mul]
exact NormOneLorentzVector.FuturePointing.metric_reflect_mem_mem h h'
lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
rw [IsOrthochronous_iff_transpose] at h
rw [IsOrthochronous_iff_futurePointing] at h h'
rw [IsOrthochronous, timeComp_mul]
exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem h h'
| lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') | LorentzGroup.mul_not_othchron_of_othchron_not_othchron | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 143,
"tokenPositionInFile": 5598,
"theoremPositionInFile": 18
} | {
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"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
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"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
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"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
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"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"HepLean.SpaceTime.LorentzGroup.Proper"
]
} | {
"hasProof": true,
"proof": ":= by\n rw [not_orthochronous_iff_le_zero, timeComp_mul]\n rw [IsOrthochronous_iff_transpose] at h\n rw [IsOrthochronous_iff_futurePointing] at h h'\n exact NormOneLorentzVector.FuturePointing.metric_reflect_mem_not_mem h h'",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 224
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
| lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ | minkowskiMetric.ge_abs_inner_product | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "fdf5fda1e73e043ba40a3836ed1f00ddcc259373",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 216,
"tokenPositionInFile": 6770,
"theoremPositionInFile": 19
} | {
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"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
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"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
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"Mathlib.Data.Set.Intervals.Infinite",
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"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
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"Mathlib.Data.Finsupp.Fin",
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"Mathlib.Algebra.Algebra.Hom",
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"Mathlib.Algebra.Module.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
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"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
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"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.LinearAlgebra.Pi",
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"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
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"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
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"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
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"Mathlib.Algebra.Ring.OrderSynonym",
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"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
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"Mathlib.SetTheory.Ordinal.Exponential",
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"Mathlib.SetTheory.Cardinal.Ordinal",
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"Mathlib.LinearAlgebra.AffineSpace.Basic",
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"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
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"Mathlib.Algebra.DirectSum.Module",
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"Mathlib.Topology.EMetricSpace.Lipschitz",
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"Mathlib.SetTheory.Cardinal.Continuum",
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"Mathlib.RingTheory.AlgebraTower",
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"Mathlib.Data.Matrix.Basic",
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"Mathlib.LinearAlgebra.Matrix.Trace",
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"Mathlib.LinearAlgebra.StdBasis",
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"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
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"Mathlib.GroupTheory.NoncommPiCoprod",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
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"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
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"Mathlib.Algebra.FreeAlgebra",
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"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
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"Mathlib.GroupTheory.Subgroup.Simple",
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"Mathlib.GroupTheory.PGroup",
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"Mathlib.LinearAlgebra.Dimension.DivisionRing",
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"Mathlib.LinearAlgebra.Dimension.LinearMap",
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"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
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"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
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"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
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"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
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"Mathlib.Data.Polynomial.Degree.Lemmas",
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"Mathlib.Data.Polynomial.Reverse",
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"Mathlib.RingTheory.EuclideanDomain",
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"Mathlib.Tactic.ComputeDegree",
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"Mathlib.Logic.Equiv.TransferInstance",
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"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
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"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
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"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
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"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
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"Mathlib.Data.Matrix.DMatrix",
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} | {
"hasProof": true,
"proof": ":= by\n rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]\n exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 105
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space time
This file introduce 4d Minkowski spacetime.
-/
noncomputable section
/-- The space-time -/
def SpaceTime : Type := Fin 4 → ℝ
/-- Give spacetime the structure of an additive commutative monoid. -/
instance : AddCommMonoid SpaceTime := Pi.addCommMonoid
/-- Give spacetime the structure of a module over the reals. -/
instance : Module ℝ SpaceTime := Pi.module _ _ _
instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normedAddCommGroup
instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
namespace SpaceTime
open Manifold
open Matrix
open Complex
open ComplexConjugate
/-- The space part of spacetime. -/
@[simp]
def space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
/-- The structure of a smooth manifold on spacetime. -/
def asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime := 𝓘(ℝ, SpaceTime)
instance : ChartedSpace SpaceTime SpaceTime := chartedSpaceSelf SpaceTime
/-- The standard basis for spacetime. -/
def stdBasis : Basis (Fin 4) ℝ SpaceTime := Pi.basisFun ℝ (Fin 4)
lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 := by
erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']
lemma stdBasis_not_eq {μ ν : Fin 4} (h : μ ≠ ν) : stdBasis μ ν = 0 := by
rw [stdBasis_apply]
exact if_neg h
lemma stdBasis_0 : stdBasis 0 = ![1, 0, 0, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_1 : stdBasis 1 = ![0, 1, 0, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_2 : stdBasis 2 = ![0, 0, 1, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_3 : stdBasis 3 = ![0, 0, 0, 1] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_mulVec (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
(Λ *ᵥ stdBasis μ) ν = Λ ν μ := by
rw [mulVec, dotProduct, Fintype.sum_eq_single μ, stdBasis_apply]
· simp only [↓reduceIte, mul_one]
· intro x h
rw [stdBasis_apply, if_neg (Ne.symm h)]
exact CommMonoidWithZero.mul_zero (Λ ν x)
lemma explicit (x : SpaceTime) : x = ![x 0, x 1, x 2, x 3] := by
funext i
fin_cases i <;> rfl
@[simp]
lemma add_apply (x y : SpaceTime) (i : Fin 4) : (x + y) i = x i + y i := rfl
| @[simp]
lemma smul_apply (x : SpaceTime) (a : ℝ) (i : Fin 4) : (a • x) i = a * x i | SpaceTime.smul_apply | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "5a5540ba782a4f6fd72fe122514f3c9c82c2b90a",
"date": "2024-04-26T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
"lineInFile": 87,
"tokenPositionInFile": 2560,
"theoremPositionInFile": 13
} | {
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"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`.
We define the give a series of lemmas related to the determinant of the Lorentz group.
-/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
open minkowskiMetric
variable {d : ℕ}
/-- The determinant of a member of the Lorentz group is `1` or `-1`. -/
lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
refine mul_self_eq_one_iff.mp ?_
simpa only [det_mul, det_dual, det_one] using congrArg det ((mem_iff_self_mul_dual).mp Λ.2)
local notation "ℤ₂" => Multiplicative (ZMod 2)
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
instance : DiscreteTopology ℤ₂ := by
exact forall_open_iff_discrete.mp fun _ => trivial
instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
instance _root_.Finite.instDiscreteTopology [TopologicalSpace X] [T1Space X] [Finite X] : DiscreteTopology X :=
discreteTopology_iff_forall_isClosed.mpr (· |>.toFinite.isClosed)
/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
@[simps!]
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
toFun x := if x = ⟨1, Set.mem_insert_of_mem (-1) rfl⟩
then Additive.toMul 0 else Additive.toMul (1 : ZMod 2)
continuous_toFun := continuous_of_discreteTopology
/-- The continuous map taking a Lorentz matrix to its determinant. -/
def detContinuous : C(𝓛 d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨Λ.1.det, Or.symm (LorentzGroup.det_eq_one_or_neg_one _)⟩,
continuous_toFun := by
refine Continuous.subtype_mk ?_ _
exact Continuous.matrix_det $
Continuous.comp' (continuous_iff_le_induced.mpr fun U a => a) continuous_id'
}
| @[simp]
lemma _root_.toMul_eq_one {α : Type*} [One α] {x : Additive α} :
Additive.toMul x = 1 ↔ x = 0 | toMul_eq_one | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "17f09022db0e73dffc3a3dff364831cc391abdb4",
"date": "2024-09-04T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
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"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
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"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
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"Mathlib.Order.SupClosed",
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"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
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"Mathlib.Data.DFinsupp.Basic",
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"Mathlib.Data.Set.Semiring",
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"Mathlib.Data.Finsupp.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
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"Mathlib.LinearAlgebra.Basis",
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"Mathlib.Data.ZMod.Basic",
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"Mathlib.RingTheory.AlgebraTower",
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"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
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"Mathlib.RingTheory.Congruence",
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"Mathlib.Data.Fintype.Perm",
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"Mathlib.RingTheory.EuclideanDomain",
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"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
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"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
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"Mathlib.LinearAlgebra.Alternating.Basic",
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"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
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"Mathlib.LinearAlgebra.TensorProduct.Tower",
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"Mathlib.RingTheory.Adjoin.FG",
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"proofType": "term",
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} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
| lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y | LorentzVector.contrDownUp_tmul_eq_dotProduct | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a65fb06605cbfe5b3907ae8a3963ce7cd664a975",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 105,
"tokenPositionInFile": 3702,
"theoremPositionInFile": 5
} | {
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"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
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"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
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"Mathlib.Topology.UniformSpace.Cauchy",
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"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
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"Mathlib.Logic.Small.Basic",
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"Mathlib.Data.Nat.PartENat",
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"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
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"Mathlib.GroupTheory.Subgroup.MulOpposite",
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"Mathlib.Data.Sigma.Lex",
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"Mathlib.GroupTheory.Subsemigroup.Membership",
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"Mathlib.RingTheory.Subsemiring.Basic",
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"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
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"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
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"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
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"Mathlib.Data.Set.Intervals.Infinite",
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"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
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"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
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"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
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"Mathlib.Order.OmegaCompletePartialOrder",
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"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
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"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
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"Mathlib.Algebra.Module.Submodule.Pointwise",
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"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
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"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
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"Mathlib.Algebra.Ring.OrderSynonym",
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"Mathlib.Data.Finsupp.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
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"Mathlib.SetTheory.Cardinal.Ordinal",
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"Mathlib.LinearAlgebra.Basis",
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"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
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"Mathlib.Data.Int.Interval",
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"Mathlib.Algebra.Order.Field.InjSurj",
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"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
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"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
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"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
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"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
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"Mathlib.Topology.Instances.Real",
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"Mathlib.Data.Complex.Abs",
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"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
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"Mathlib.Topology.EMetricSpace.Lipschitz",
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"Mathlib.SetTheory.Cardinal.Continuum",
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"Mathlib.RingTheory.AlgebraTower",
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"Mathlib.LinearAlgebra.StdBasis",
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"Mathlib.RingTheory.Finiteness",
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"Mathlib.RingTheory.UniqueFactorizationDomain",
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"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
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"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
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"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
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"Mathlib.Data.Fintype.Perm",
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"Mathlib.GroupTheory.Perm.Closure",
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"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
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"Mathlib.Algebra.FreeAlgebra",
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"Mathlib.Algebra.CharP.ExpChar",
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"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
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"Mathlib.GroupTheory.Subgroup.Simple",
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"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
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"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
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"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
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"Mathlib.Data.Nat.WithBot",
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"Mathlib.Data.Polynomial.Monomial",
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"Mathlib.Data.Nat.Cast.WithTop",
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"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
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"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
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"Mathlib.Data.Polynomial.Div",
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"Mathlib.Tactic.ComputeDegree",
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"Mathlib.RingTheory.Ideal.LocalRing",
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"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
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"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
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"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
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"Mathlib.LinearAlgebra.TensorProduct.Tower",
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"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
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"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic",
"HepLean.SpaceTime.LorentzVector.LorentzAction",
"HepLean.SpaceTime.LorentzVector.Covariant",
"Mathlib.LinearAlgebra.Multilinear.TensorProduct",
"Mathlib.LinearAlgebra.PiTensorProduct",
"HepLean.Tensors.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n rw [dotProduct_comm x y]\n rfl",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 38
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
simp only [unitUp, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
/-- The unit of the contraction of covariant Lorentz vector and a
contravariant Lorentz vector. -/
def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x := by
simp only [unitDown, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,
lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']
/-!
# Contractions and the Lorentz actions
-/
open Matrix
@[simp]
lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) := by
rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
@[simp]
lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
(LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) := by
rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]
rw [dotProduct_comm, dotProduct_comm x y]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
end LorentzVector
namespace minkowskiMatrix
open LorentzVector
open TensorStructure
open scoped minkowskiMatrix
variable {d : ℕ}
/-- The metric tensor as an element of `LorentzVector d ⊗[ℝ] LorentzVector d`. -/
def asTenProd : LorentzVector d ⊗[ℝ] LorentzVector d :=
∑ μ, ∑ ν, η μ ν • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis ν)
lemma asTenProd_diag :
@asTenProd d = ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ) := by
simp only [asTenProd]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [Finset.sum_eq_single μ]
· intro ν _ hμν
rw [minkowskiMatrix.off_diag_zero hμν.symm]
exact TensorProduct.zero_smul (e μ ⊗ₜ[ℝ] e ν)
· intro a
rename_i j
exact False.elim (a j)
/-- The metric tensor as an element of `CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d`. -/
def asCoTenProd : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ μ, ∑ ν, η μ ν • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν)
lemma asCoTenProd_diag :
@asCoTenProd d = ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ]
CovariantLorentzVector.stdBasis μ) := by
simp only [asCoTenProd]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [Finset.sum_eq_single μ]
· intro ν _ hμν
rw [minkowskiMatrix.off_diag_zero hμν.symm]
simp only [zero_smul]
· intro a
simp_all only
@[simp]
lemma contrLeft_asTenProd (x : CovariantLorentzVector d) :
contrLeftAux contrDownUp (x ⊗ₜ[ℝ] asTenProd) =
∑ μ, ((η μ μ * x μ) • LorentzVector.stdBasis μ) := by
simp only [asTenProd_diag]
rw [tmul_sum]
rw [map_sum]
refine Finset.sum_congr rfl (fun μ _ => ?_)
simp only [contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq, lid_tmul]
exact smul_smul (η μ μ) (x μ) (e μ)
@[simp]
lemma contrLeft_asCoTenProd (x : LorentzVector d) :
contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asCoTenProd) =
∑ μ, ((η μ μ * x μ) • CovariantLorentzVector.stdBasis μ) := by
simp only [asCoTenProd_diag]
rw [tmul_sum]
rw [map_sum]
refine Finset.sum_congr rfl (fun μ _ => ?_)
simp only [contrLeftAux, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul]
exact smul_smul (η μ μ) (x μ) (CovariantLorentzVector.stdBasis μ)
@[simp]
lemma asTenProd_contr_asCoTenProd :
(contrMidAux (contrUpDown) (asTenProd ⊗ₜ[ℝ] asCoTenProd))
= TensorProduct.comm ℝ _ _ (@unitUp d) := by
simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asTenProd_diag, LinearMap.coe_comp,
LinearEquiv.coe_coe, Function.comp_apply]
rw [sum_tmul, map_sum, map_sum, unitUp]
simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, comm_tmul, map_sum]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [← tmul_smul, TensorProduct.assoc_tmul]
simp only [map_tmul, LinearMap.id_coe, id_eq, contrLeft_asCoTenProd]
rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul]
· change (η μ μ * (η μ μ * e μ μ)) • e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ = _
rw [LorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one, η_apply_mul_η_apply_diag, one_smul]
· exact Finset.mem_univ μ
· intro ν _ hμν
rw [tmul_smul]
change (η ν ν * (η μ μ * e μ ν)) • (e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν) = _
rw [LorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', mul_ite, mul_one, mul_zero, ite_smul, zero_smul,
ite_eq_right_iff, smul_eq_zero, mul_eq_zero]
exact fun a => False.elim (hμν a.symm)
@[simp]
lemma asCoTenProd_contr_asTenProd :
(contrMidAux (contrDownUp) (asCoTenProd ⊗ₜ[ℝ] asTenProd))
= TensorProduct.comm ℝ _ _ (@unitDown d) := by
simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asCoTenProd_diag,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
rw [sum_tmul, map_sum, map_sum, unitDown]
simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, comm_tmul, map_sum]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [smul_tmul, TensorProduct.assoc_tmul]
simp only [tmul_smul, map_smul, map_tmul, LinearMap.id_coe, id_eq,
contrLeft_asTenProd]
rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul, smul_smul, LorentzVector.stdBasis]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one, η_apply_mul_η_apply_diag, one_smul]
· exact Finset.mem_univ μ
· intro ν _ hμν
rw [tmul_smul]
rw [LorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', mul_ite, mul_one, mul_zero, ite_smul, zero_smul,
ite_eq_right_iff, smul_eq_zero]
exact fun a => False.elim (hμν a.symm)
| lemma asTenProd_invariant (g : LorentzGroup d) :
TensorProduct.map (LorentzVector.rep g) (LorentzVector.rep g) asTenProd = asTenProd | minkowskiMatrix.asTenProd_invariant | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 287,
"tokenPositionInFile": 11541,
"theoremPositionInFile": 20
} | {
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"numInFilePremises": 1,
"repositoryPremises": true,
"numRepositoryPremises": 13,
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"Lean.Meta.Tactic.AC",
"Lean.Meta.Tactic",
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"Lean.Util",
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"Lean.LoadDynlib",
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"Lean.Util.LakePath",
"Lean.Server.FileWorker.SetupFile",
"Lean.Server.ImportCompletion",
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"Lean.Server.Rpc.Deriving",
"Lean.Server.Rpc",
"Lean.Server",
"Lean.Widget",
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"Lean.Linter",
"Lean",
"Std.Classes.BEq",
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"Std.Control.Lemmas",
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"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
"Std.Data.Bool",
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"Std.Data.List.Lemmas",
"Std.Tactic.SeqFocus",
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"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
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"Std.Data.BinomialHeap",
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"Std.Data.Int.Gcd",
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"Std.Data.List.Pairwise",
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"Std.Data.List",
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"Std.Data.MLList.IO",
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"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
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"Std.Data.Range",
"Std.Data.Rat.Lemmas",
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"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
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"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
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"Std.Lean.Delaborator",
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"Std.Lean.Float",
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"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
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"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
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"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic",
"HepLean.SpaceTime.LorentzVector.LorentzAction",
"HepLean.SpaceTime.LorentzVector.Covariant",
"Mathlib.LinearAlgebra.Multilinear.TensorProduct",
"Mathlib.LinearAlgebra.PiTensorProduct",
"HepLean.Tensors.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [asTenProd, map_sum, map_smul, map_tmul, rep_apply_stdBasis,\n Matrix.transpose_apply]\n trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d,\n η μ ν • (g.1 φ μ • e φ) ⊗ₜ[ℝ] ∑ ρ : Fin 1 ⊕ Fin d, g.1 ρ ν • e ρ\n · refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))\n rw [sum_tmul, Finset.smul_sum]\n trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,\n (η μ ν • (g.1 φ μ • e φ) ⊗ₜ[ℝ] (g.1 ρ ν • e ρ))\n · refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ =>\n Finset.sum_congr rfl (fun φ _ => ?_)))\n rw [tmul_sum, Finset.smul_sum]\n rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_comm)]\n rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => Finset.sum_comm))]\n rw [Finset.sum_comm]\n rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_comm)]\n trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d, ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d,\n ((g.1 φ μ * η μ ν * g.1 ρ ν) • (e φ) ⊗ₜ[ℝ] (e ρ))\n · refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>\n Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))))\n rw [smul_tmul, tmul_smul, tmul_smul, smul_smul, smul_smul]\n ring_nf\n rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>\n Finset.sum_congr rfl (fun μ _ => Finset.sum_smul.symm)))]\n rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => Finset.sum_smul.symm))]\n trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,\n (((g.1 * minkowskiMatrix * g.1.transpose : Matrix _ _ _) φ ρ) • (e φ) ⊗ₜ[ℝ] (e ρ))\n · refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => ?_))\n apply congrFun\n apply congrArg\n simp only [Matrix.mul_apply, Matrix.transpose_apply]\n rw [Finset.sum_comm]\n refine Finset.sum_congr rfl (fun μ _ => ?_)\n rw [Finset.sum_mul]\n simp",
"proofType": "tactic",
"proofLengthLines": 34,
"proofLengthTokens": 1921
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
| lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ | minkowskiMetric.eq_time_minus_inner_prod | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 150,
"tokenPositionInFile": 4670,
"theoremPositionInFile": 11
} | {
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"Mathlib.Tactic.Says",
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"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
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"Mathlib.Tactic.UnsetOption",
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"ProofWidgets.Component.Basic",
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"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
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"Mathlib.Data.Set.Image",
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"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
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"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
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"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical"
]
} | {
"hasProof": true,
"proof": ":= by\n rw [as_sum, @PiLp.inner_apply]\n rfl",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 44
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
@[simp]
lemma dual_id : @dual d 1 = 1 := by
simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
@[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
simp only [dual, transpose_mul]
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
· noncomm_ring [minkowskiMatrix.sq]
· noncomm_ring
@[simp]
lemma dual_dual : dual (dual Λ) = Λ := by
simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
trans (η * η) * Λ * (η * η)
· noncomm_ring
· noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_eta : @dual d η = η := by
simp only [dual, eq_transpose]
noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
noncomm_ring
| @[simp]
lemma det_dual : (dual Λ).det = Λ.det | minkowskiMetric.det_dual | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
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"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
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"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
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"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
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"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
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"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
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} | {
"hasProof": true,
"proof": ":= by\n simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]\n group\n norm_cast\n simp",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 113
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
/-!
# Lorentz vector as a self-adjoint matrix
There is a linear equivalence between the vector space of space-time points
and the vector space of 2×2-complex self-adjoint matrices.
In this file we define this linear equivalence in `toSelfAdjointMatrix`.
-/
/-! TODO: Generalize rep of Lorentz vector as a self-adjoint matrix to arbitrary dimension. -/
namespace SpaceTime
open Matrix
open MatrixGroups
open Complex
/-- A 2×2-complex matrix formed from a space-time point. -/
@[simp]
def toMatrix (x : LorentzVector 3) : Matrix (Fin 2) (Fin 2) ℂ :=
!![x.time + x.space 2, x.space 0 - x.space 1 * I; x.space 0 + x.space 1 * I, x.time - x.space 2]
| /-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/
lemma toMatrix_isSelfAdjoint (x : LorentzVector 3) : IsSelfAdjoint (toMatrix x) | SpaceTime.toMatrix_isSelfAdjoint | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | {
"commit": "ea4327aff59fde778912eb081ed0847eb114af44",
"date": "2024-06-12T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/AsSelfAdjointMatrix.lean | HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix | HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix.jsonl | {
"lineInFile": 29,
"tokenPositionInFile": 911,
"theoremPositionInFile": 1
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"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
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"Mathlib.Data.Set.Pointwise.ListOfFn",
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"Mathlib.Data.Set.Semiring",
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"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
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"Mathlib.RingTheory.Ideal.LocalRing",
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"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
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"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
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"Mathlib.FieldTheory.Tower",
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"Mathlib.GroupTheory.Archimedean",
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} | {
"hasProof": true,
"proof": ":= by\n rw [isSelfAdjoint_iff, star_eq_conjTranspose, ← Matrix.ext_iff]\n intro i j\n fin_cases i <;> fin_cases j <;>\n simp [toMatrix, conj_ofReal]\n rfl",
"proofType": "tactic",
"proofLengthLines": 5,
"proofLengthTokens": 156
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
lemma IsOrthochronous_iff_ge_one :
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
NormOneLorentzVector.time_pos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_neg_one :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ -1 := by
rw [timeComp, IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.not_mem_iff,
NormOneLorentzVector.time_nonpos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_zero :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [not_orthochronous_iff_le_neg_one] at h
linarith
· rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
def stepFunction : ℝ → ℝ := fun t =>
if t ≤ -1 then -1 else
if 1 ≤ t then 1 else t
lemma stepFunction_continuous : Continuous stepFunction := by
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
<;> intro a ha
· rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp only [le_neg_self_iff, id_eq]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
· rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
exact id (Eq.symm ha)
/-- The continuous map from `lorentzGroup` to `ℝ` wh
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
def orthchroMapReal : C(LorentzGroup d, ℝ) := ContinuousMap.comp
⟨stepFunction, stepFunction_continuous⟩ timeCompCont
lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMapReal Λ = 1 := by
rw [IsOrthochronous_iff_ge_one, timeComp] at h
change stepFunction (Λ.1 _ _) = 1
rw [stepFunction, if_pos h, if_neg]
linarith
lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMapReal Λ = - 1 := by
rw [not_orthochronous_iff_le_neg_one] at h
change stepFunction (timeComp _)= - 1
rw [stepFunction, if_pos h]
lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
by_cases h : IsOrthochronous Λ
· exact Or.inr $ orthchroMapReal_on_IsOrthochronous h
· exact Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
local notation "ℤ₂" => Multiplicative (ZMod 2)
/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
def orthchroMap : C(LorentzGroup d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMap Λ = 1 := by
simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
simp only [orthchroMap, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
orthchroMapReal_on_not_IsOrthochronous h, coeForℤ₂_apply, Subtype.mk.injEq, Nat.reduceAdd]
rw [if_neg]
· rfl
· linarith
lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
(h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
rw [IsOrthochronous_iff_transpose] at h
rw [IsOrthochronous_iff_futurePointing] at h h'
rw [IsOrthochronous, timeComp_mul]
exact NormOneLorentzVector.FuturePointing.metric_reflect_mem_mem h h'
lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
rw [IsOrthochronous_iff_transpose] at h
rw [IsOrthochronous_iff_futurePointing] at h h'
rw [IsOrthochronous, timeComp_mul]
exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem h h'
lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
rw [not_orthochronous_iff_le_zero, timeComp_mul]
rw [IsOrthochronous_iff_transpose] at h
rw [IsOrthochronous_iff_futurePointing] at h h'
exact NormOneLorentzVector.FuturePointing.metric_reflect_mem_not_mem h h'
| lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
(h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') | LorentzGroup.mul_not_othchron_of_not_othchron_othchron | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
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"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
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"Mathlib.Data.Fintype.BigOperators",
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"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
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"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
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"Mathlib.Topology.Algebra.MulAction",
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"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
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"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
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"Mathlib.GroupTheory.Subgroup.ZPowers",
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"Mathlib.Data.ZMod.Defs",
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"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
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"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
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"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
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"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
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"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
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"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
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"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
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"Mathlib.Algebra.Module.Submodule.LinearMap",
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"Mathlib.Data.List.NatAntidiagonal",
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"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
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"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
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"Mathlib.Algebra.Module.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
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"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
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"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
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"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
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"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
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"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
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"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
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"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
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"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
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"Mathlib.Topology.Order.MonotoneContinuity",
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"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
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"Mathlib.LinearAlgebra.GeneralLinearGroup",
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"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
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"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
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"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
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"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
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"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
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"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
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"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
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"Mathlib.Algebra.Algebra.Pi",
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"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
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"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
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"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
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"Mathlib.Data.PEquiv",
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"Mathlib.LinearAlgebra.Alternating.Basic",
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"Mathlib.LinearAlgebra.TensorProduct.Tower",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
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"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
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} | {
"hasProof": true,
"proof": ":= by\n rw [not_orthochronous_iff_le_zero, timeComp_mul]\n rw [IsOrthochronous_iff_transpose] at h\n rw [IsOrthochronous_iff_futurePointing] at h h'\n exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_mem h h'",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 224
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import HepLean.SpaceTime.LorentzGroup.Basic
import Mathlib.RepresentationTheory.Basic
/-!
# Lorentz group action on Lorentz vectors.
-/
noncomputable section
open BigOperators
namespace LorentzVector
variable {d : ℕ} (v : LorentzVector d)
/-- The contravariant action of the Lorentz group on a Lorentz vector. -/
def rep : Representation ℝ (LorentzGroup d) (LorentzVector d) where
toFun g := Matrix.toLinAlgEquiv e g
map_one' := (MulEquivClass.map_eq_one_iff (Matrix.toLinAlgEquiv e)).mpr rfl
map_mul' x y := by
simp only [lorentzGroupIsGroup_mul_coe, map_mul]
open Matrix in
lemma rep_apply (g : LorentzGroup d) : rep g v = g *ᵥ v := rfl
| lemma rep_apply_stdBasis (g : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :
rep g (stdBasis μ) = ∑ ν, g.1.transpose μ ν • stdBasis ν | LorentzVector.rep_apply_stdBasis | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/LorentzAction.lean | HepLean.SpaceTime.LorentzVector.LorentzAction | HepLean.SpaceTime.LorentzVector.LorentzAction.jsonl | {
"lineInFile": 32,
"tokenPositionInFile": 864,
"theoremPositionInFile": 1
} | {
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"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
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"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
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"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
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"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
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"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
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"Mathlib.Data.ZMod.Defs",
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"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
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"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
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"Mathlib.Order.Hom.Order",
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"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
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"Mathlib.Data.Nat.PartENat",
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"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
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"Mathlib.Data.Nat.Prime",
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"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
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"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
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"Mathlib.Data.Nat.Cast.Prod",
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"Mathlib.GroupTheory.Subsemigroup.Membership",
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"Mathlib.RingTheory.Subsemiring.Basic",
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"Mathlib.Order.Filter.Extr",
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"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
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"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
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"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
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"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
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"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
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"Mathlib.Topology.Algebra.Star",
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"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
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"Mathlib.Data.Finsupp.Indicator",
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"Mathlib.Algebra.Module.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
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"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
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"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
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"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
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"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
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"Mathlib.Algebra.Ring.OrderSynonym",
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"Mathlib.Data.Finsupp.Order",
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"Mathlib.Data.Sum.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
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"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
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"Mathlib.LinearAlgebra.Basis",
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"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
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"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
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"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
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"Mathlib.Algebra.Order.Module.Pointwise",
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"Mathlib.Data.Real.Pointwise",
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"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
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"Mathlib.Data.ENNReal.Inv",
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"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
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"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
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"Mathlib.Algebra.Algebra.Subalgebra.Basic",
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"Mathlib.Algebra.DirectSum.Finsupp",
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"Mathlib.LinearAlgebra.Matrix.Trace",
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"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
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"Mathlib.RingTheory.Congruence",
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"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
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"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
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"Mathlib.GroupTheory.NoncommPiCoprod",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
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"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
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"Mathlib.GroupTheory.Perm.Closure",
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"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
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"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
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"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
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"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
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"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
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"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
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"Mathlib.Data.Polynomial.Reverse",
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"Mathlib.RingTheory.EuclideanDomain",
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"Mathlib.Tactic.ComputeDegree",
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"Mathlib.RingTheory.Ideal.LocalRing",
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"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
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"Mathlib.Data.Fintype.Sort",
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"Mathlib.FieldTheory.Tower",
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"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
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} | {
"hasProof": true,
"proof": ":= by\n simp only [rep_apply, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,\n Finset.sum_singleton, decomp_stdBasis']\n funext ν\n simp only [stdBasis, Matrix.transpose_apply]\n erw [Pi.basisFun_apply, Matrix.mulVec_single_one]\n rfl",
"proofType": "tactic",
"proofLengthLines": 6,
"proofLengthTokens": 265
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
| lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d | LorentzGroup.IsOrthochronous_iff_futurePointing | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 33,
"tokenPositionInFile": 791,
"theoremPositionInFile": 1
} | {
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"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
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"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"HepLean.SpaceTime.LorentzGroup.Proper"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]\n rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 134
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
simp only [unitUp, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
/-- The unit of the contraction of covariant Lorentz vector and a
contravariant Lorentz vector. -/
def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x := by
simp only [unitDown, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,
lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']
/-!
# Contractions and the Lorentz actions
-/
open Matrix
@[simp]
lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) := by
rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
@[simp]
lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
(LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) := by
rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]
rw [dotProduct_comm, dotProduct_comm x y]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
end LorentzVector
namespace minkowskiMatrix
open LorentzVector
open TensorStructure
open scoped minkowskiMatrix
variable {d : ℕ}
/-- The metric tensor as an element of `LorentzVector d ⊗[ℝ] LorentzVector d`. -/
def asTenProd : LorentzVector d ⊗[ℝ] LorentzVector d :=
∑ μ, ∑ ν, η μ ν • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis ν)
lemma asTenProd_diag :
@asTenProd d = ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ) := by
simp only [asTenProd]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [Finset.sum_eq_single μ]
· intro ν _ hμν
rw [minkowskiMatrix.off_diag_zero hμν.symm]
exact TensorProduct.zero_smul (e μ ⊗ₜ[ℝ] e ν)
· intro a
rename_i j
exact False.elim (a j)
/-- The metric tensor as an element of `CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d`. -/
def asCoTenProd : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ μ, ∑ ν, η μ ν • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν)
lemma asCoTenProd_diag :
@asCoTenProd d = ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ]
CovariantLorentzVector.stdBasis μ) := by
simp only [asCoTenProd]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [Finset.sum_eq_single μ]
· intro ν _ hμν
rw [minkowskiMatrix.off_diag_zero hμν.symm]
simp only [zero_smul]
· intro a
simp_all only
@[simp]
lemma contrLeft_asTenProd (x : CovariantLorentzVector d) :
contrLeftAux contrDownUp (x ⊗ₜ[ℝ] asTenProd) =
∑ μ, ((η μ μ * x μ) • LorentzVector.stdBasis μ) := by
simp only [asTenProd_diag]
rw [tmul_sum]
rw [map_sum]
refine Finset.sum_congr rfl (fun μ _ => ?_)
simp only [contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq, lid_tmul]
exact smul_smul (η μ μ) (x μ) (e μ)
@[simp]
lemma contrLeft_asCoTenProd (x : LorentzVector d) :
contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asCoTenProd) =
∑ μ, ((η μ μ * x μ) • CovariantLorentzVector.stdBasis μ) := by
simp only [asCoTenProd_diag]
rw [tmul_sum]
rw [map_sum]
refine Finset.sum_congr rfl (fun μ _ => ?_)
simp only [contrLeftAux, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul]
exact smul_smul (η μ μ) (x μ) (CovariantLorentzVector.stdBasis μ)
@[simp]
lemma asTenProd_contr_asCoTenProd :
(contrMidAux (contrUpDown) (asTenProd ⊗ₜ[ℝ] asCoTenProd))
= TensorProduct.comm ℝ _ _ (@unitUp d) := by
simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asTenProd_diag, LinearMap.coe_comp,
LinearEquiv.coe_coe, Function.comp_apply]
rw [sum_tmul, map_sum, map_sum, unitUp]
simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, comm_tmul, map_sum]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [← tmul_smul, TensorProduct.assoc_tmul]
simp only [map_tmul, LinearMap.id_coe, id_eq, contrLeft_asCoTenProd]
rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul]
· change (η μ μ * (η μ μ * e μ μ)) • e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ = _
rw [LorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one, η_apply_mul_η_apply_diag, one_smul]
· exact Finset.mem_univ μ
· intro ν _ hμν
rw [tmul_smul]
change (η ν ν * (η μ μ * e μ ν)) • (e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν) = _
rw [LorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', mul_ite, mul_one, mul_zero, ite_smul, zero_smul,
ite_eq_right_iff, smul_eq_zero, mul_eq_zero]
exact fun a => False.elim (hμν a.symm)
| @[simp]
lemma asCoTenProd_contr_asTenProd :
(contrMidAux (contrDownUp) (asCoTenProd ⊗ₜ[ℝ] asTenProd))
= TensorProduct.comm ℝ _ _ (@unitDown d) | minkowskiMatrix.asCoTenProd_contr_asTenProd | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 262,
"tokenPositionInFile": 10377,
"theoremPositionInFile": 19
} | {
"inFilePremises": true,
"numInFilePremises": 6,
"repositoryPremises": true,
"numRepositoryPremises": 20,
"numPremises": 200,
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"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic",
"HepLean.SpaceTime.LorentzVector.LorentzAction",
"HepLean.SpaceTime.LorentzVector.Covariant",
"Mathlib.LinearAlgebra.Multilinear.TensorProduct",
"Mathlib.LinearAlgebra.PiTensorProduct",
"HepLean.Tensors.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asCoTenProd_diag,\n LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]\n rw [sum_tmul, map_sum, map_sum, unitDown]\n simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,\n Finset.sum_singleton, map_add, comm_tmul, map_sum]\n refine Finset.sum_congr rfl (fun μ _ => ?_)\n rw [smul_tmul, TensorProduct.assoc_tmul]\n simp only [tmul_smul, map_smul, map_tmul, LinearMap.id_coe, id_eq,\n contrLeft_asTenProd]\n rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul, smul_smul, LorentzVector.stdBasis]\n · erw [Pi.basisFun_apply]\n simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one, η_apply_mul_η_apply_diag, one_smul]\n · exact Finset.mem_univ μ\n · intro ν _ hμν\n rw [tmul_smul]\n rw [LorentzVector.stdBasis]\n erw [Pi.basisFun_apply]\n simp only [LinearMap.stdBasis_apply', mul_ite, mul_one, mul_zero, ite_smul, zero_smul,\n ite_eq_right_iff, smul_eq_zero]\n exact fun a => False.elim (hμν a.symm)",
"proofType": "tactic",
"proofLengthLines": 20,
"proofLengthTokens": 1011
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
@[simp]
lemma dual_id : @dual d 1 = 1 := by
simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
@[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
simp only [dual, transpose_mul]
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
· noncomm_ring [minkowskiMatrix.sq]
· noncomm_ring
@[simp]
lemma dual_dual : dual (dual Λ) = Λ := by
simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
trans (η * η) * Λ * (η * η)
· noncomm_ring
· noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_eta : @dual d η = η := by
simp only [dual, eq_transpose]
noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
noncomm_ring
@[simp]
lemma det_dual : (dual Λ).det = Λ.det := by
simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
group
norm_cast
simp
| @[simp]
lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ | minkowskiMetric.dual_mulVec_right | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 272,
"tokenPositionInFile": 8332,
"theoremPositionInFile": 28
} | {
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"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
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"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,\n mulVec_mulVec, ← mul_assoc, minkowskiMatrix.sq, one_mul, (vecMul_transpose Λ x).symm, ←\n dotProduct_mulVec]",
"proofType": "tactic",
"proofLengthLines": 3,
"proofLengthTokens": 216
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
/-!
# Lorentz vector as a self-adjoint matrix
There is a linear equivalence between the vector space of space-time points
and the vector space of 2×2-complex self-adjoint matrices.
In this file we define this linear equivalence in `toSelfAdjointMatrix`.
-/
/-! TODO: Generalize rep of Lorentz vector as a self-adjoint matrix to arbitrary dimension. -/
namespace SpaceTime
open Matrix
open MatrixGroups
open Complex
/-- A 2×2-complex matrix formed from a space-time point. -/
@[simp]
def toMatrix (x : LorentzVector 3) : Matrix (Fin 2) (Fin 2) ℂ :=
!![x.time + x.space 2, x.space 0 - x.space 1 * I; x.space 0 + x.space 1 * I, x.time - x.space 2]
/-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/
lemma toMatrix_isSelfAdjoint (x : LorentzVector 3) : IsSelfAdjoint (toMatrix x) := by
rw [isSelfAdjoint_iff, star_eq_conjTranspose, ← Matrix.ext_iff]
intro i j
fin_cases i <;> fin_cases j <;>
simp [toMatrix, conj_ofReal]
rfl
/-- A self-adjoint matrix formed from a space-time point. -/
@[simps!]
def toSelfAdjointMatrix' (x : LorentzVector 3) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
⟨toMatrix x, toMatrix_isSelfAdjoint x⟩
/-- A self-adjoint matrix formed from a space-time point. -/
@[simp]
noncomputable def fromSelfAdjointMatrix' (x : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
LorentzVector 3 := fun i =>
match i with
| Sum.inl 0 => 1/2 * (x.1 0 0 + x.1 1 1).re
| Sum.inr 0 => (x.1 1 0).re
| Sum.inr 1 => (x.1 1 0).im
| Sum.inr 2 => 1/2 * (x.1 0 0 - x.1 1 1).re
/-- The linear equivalence between the vector-space `spaceTime` and self-adjoint
2×2-complex matrices. -/
noncomputable def toSelfAdjointMatrix :
LorentzVector 3 ≃ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
toFun := toSelfAdjointMatrix'
invFun := fromSelfAdjointMatrix'
left_inv x := by
funext i
match i with
| Sum.inl 0 =>
simp only [fromSelfAdjointMatrix', one_div, toSelfAdjointMatrix', toMatrix,
LorentzVector.time, Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply,
cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, head_cons,
head_fin_const, add_add_sub_cancel, add_re, ofReal_re]
ring_nf
| Sum.inr 0 =>
simp [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, toMatrix_isSelfAdjoint]
| Sum.inr 1 =>
simp only [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, LorentzVector.time,
Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply, cons_val', cons_val_zero,
empty_val', cons_val_fin_one, cons_val_one, head_fin_const, add_im, ofReal_im, mul_im,
ofReal_re, I_im, mul_one, I_re, mul_zero, add_zero, zero_add]
| Sum.inr 2 =>
simp only [fromSelfAdjointMatrix', one_div, toSelfAdjointMatrix', toMatrix,
LorentzVector.time, Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply,
cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, head_cons,
head_fin_const, add_sub_sub_cancel, add_re, ofReal_re]
ring
right_inv x := by
simp only [toSelfAdjointMatrix', toMatrix, LorentzVector.time, fromSelfAdjointMatrix', one_div,
Fin.isValue, add_re, ofReal_mul, ofReal_inv, ofReal_ofNat, ofReal_add, LorentzVector.space,
Function.comp_apply, sub_re, ofReal_sub, re_add_im]
ext i j
fin_cases i <;> fin_cases j <;>
field_simp [fromSelfAdjointMatrix', toMatrix, conj_ofReal]
· exact conj_eq_iff_re.mp (congrArg (fun M => M 0 0) $ selfAdjoint.mem_iff.mp x.2)
· have h01 := congrArg (fun M => M 0 1) $ selfAdjoint.mem_iff.mp x.2
simp only [Fin.isValue, star_apply, RCLike.star_def] at h01
rw [← h01, RCLike.conj_eq_re_sub_im]
rfl
· exact conj_eq_iff_re.mp (congrArg (fun M => M 1 1) $ selfAdjoint.mem_iff.mp x.2)
map_add' x y := by
ext i j : 2
simp only [toSelfAdjointMatrix'_coe, add_apply, ofReal_add, of_apply, cons_val', empty_val',
cons_val_fin_one, AddSubmonoid.coe_add, AddSubgroup.coe_toAddSubmonoid, Matrix.add_apply]
fin_cases i <;> fin_cases j
· rw [show (x + y) (Sum.inl 0) = x (Sum.inl 0) + y (Sum.inl 0) from rfl]
rw [show (x + y) (Sum.inr 2) = x (Sum.inr 2) + y (Sum.inr 2) from rfl]
simp only [Fin.isValue, ofReal_add, Fin.zero_eta, cons_val_zero]
ring
· rw [show (x + y) (Sum.inr 0) = x (Sum.inr 0) + y (Sum.inr 0) from rfl]
rw [show (x + y) (Sum.inr 1) = x (Sum.inr 1) + y (Sum.inr 1) from rfl]
simp only [Fin.isValue, ofReal_add, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
cons_val_zero]
ring
· rw [show (x + y) (Sum.inr 0) = x (Sum.inr 0) + y (Sum.inr 0) from rfl]
rw [show (x + y) (Sum.inr 1) = x (Sum.inr 1) + y (Sum.inr 1) from rfl]
simp only [Fin.isValue, ofReal_add, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
head_fin_const]
ring
· rw [show (x + y) (Sum.inl 0) = x (Sum.inl 0) + y (Sum.inl 0) from rfl]
rw [show (x + y) (Sum.inr 2) = x (Sum.inr 2) + y (Sum.inr 2) from rfl]
simp only [Fin.isValue, ofReal_add, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
ring
map_smul' r x := by
ext i j : 2
simp only [toSelfAdjointMatrix'_coe, Fin.isValue, of_apply, cons_val', empty_val',
cons_val_fin_one, RingHom.id_apply, selfAdjoint.val_smul, smul_apply, real_smul]
fin_cases i <;> fin_cases j
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero]
ring
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
cons_val_zero]
ring
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
head_fin_const]
ring
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
ring
open minkowskiMetric in
| lemma det_eq_ηLin (x : LorentzVector 3) : det (toSelfAdjointMatrix x).1 = ⟪x, x⟫ₘ | SpaceTime.det_eq_ηLin | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | {
"commit": "de89fd7ef0ab94b46123a4a24ff680e72d957dbf",
"date": "2024-06-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/AsSelfAdjointMatrix.lean | HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix | HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix.jsonl | {
"lineInFile": 141,
"tokenPositionInFile": 6633,
"theoremPositionInFile": 4
} | {
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"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
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"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [toSelfAdjointMatrix, LinearEquiv.coe_mk, toSelfAdjointMatrix'_coe, Fin.isValue,\n det_fin_two_of, eq_time_minus_inner_prod, LorentzVector.time, LorentzVector.space,\n PiLp.inner_apply, Function.comp_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_three,\n ofReal_sub, ofReal_mul, ofReal_add]\n ring_nf\n simp only [Fin.isValue, I_sq, mul_neg, mul_one]\n ring",
"proofType": "tactic",
"proofLengthLines": 7,
"proofLengthTokens": 389
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
simp only [unitUp, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
/-- The unit of the contraction of covariant Lorentz vector and a
contravariant Lorentz vector. -/
def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x := by
simp only [unitDown, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,
lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']
/-!
# Contractions and the Lorentz actions
-/
open Matrix
@[simp]
lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) := by
rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
@[simp]
lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
(LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) := by
rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]
rw [dotProduct_comm, dotProduct_comm x y]
simp only [rep_apply, CovariantLorentzVector.rep_apply]
rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
end LorentzVector
namespace minkowskiMatrix
open LorentzVector
open TensorStructure
open scoped minkowskiMatrix
variable {d : ℕ}
/-- The metric tensor as an element of `LorentzVector d ⊗[ℝ] LorentzVector d`. -/
def asTenProd : LorentzVector d ⊗[ℝ] LorentzVector d :=
∑ μ, ∑ ν, η μ ν • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis ν)
lemma asTenProd_diag :
@asTenProd d = ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ) := by
simp only [asTenProd]
refine Finset.sum_congr rfl (fun μ _ => ?_)
rw [Finset.sum_eq_single μ]
· intro ν _ hμν
rw [minkowskiMatrix.off_diag_zero hμν.symm]
exact TensorProduct.zero_smul (e μ ⊗ₜ[ℝ] e ν)
· intro a
rename_i j
exact False.elim (a j)
/-- The metric tensor as an element of `CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d`. -/
def asCoTenProd : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ μ, ∑ ν, η μ ν • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν)
| lemma asCoTenProd_diag :
@asCoTenProd d = ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ]
CovariantLorentzVector.stdBasis μ) | minkowskiMatrix.asCoTenProd_diag | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 197,
"tokenPositionInFile": 7505,
"theoremPositionInFile": 15
} | {
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"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic",
"HepLean.SpaceTime.LorentzVector.LorentzAction",
"HepLean.SpaceTime.LorentzVector.Covariant",
"Mathlib.LinearAlgebra.Multilinear.TensorProduct",
"Mathlib.LinearAlgebra.PiTensorProduct",
"HepLean.Tensors.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [asCoTenProd]\n refine Finset.sum_congr rfl (fun μ _ => ?_)\n rw [Finset.sum_eq_single μ]\n · intro ν _ hμν\n rw [minkowskiMatrix.off_diag_zero hμν.symm]\n simp only [zero_smul]\n · intro a\n simp_all only",
"proofType": "tactic",
"proofLengthLines": 8,
"proofLengthTokens": 229
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
@[simp]
lemma dual_id : @dual d 1 = 1 := by
simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
@[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
simp only [dual, transpose_mul]
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
· noncomm_ring [minkowskiMatrix.sq]
· noncomm_ring
@[simp]
lemma dual_dual : dual (dual Λ) = Λ := by
simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
trans (η * η) * Λ * (η * η)
· noncomm_ring
· noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_eta : @dual d η = η := by
simp only [dual, eq_transpose]
noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
noncomm_ring
@[simp]
lemma det_dual : (dual Λ).det = Λ.det := by
simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
group
norm_cast
simp
@[simp]
lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,
mulVec_mulVec, ← mul_assoc, minkowskiMatrix.sq, one_mul, (vecMul_transpose Λ x).symm, ←
dotProduct_mulVec]
@[simp]
lemma dual_mulVec_left : ⟪(dual Λ) *ᵥ x, y⟫ₘ = ⟪x, Λ *ᵥ y⟫ₘ := by
rw [symm, dual_mulVec_right, symm]
lemma matrix_apply_eq_iff_sub : ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ ↔ ⟪v, (Λ - Λ') *ᵥ w⟫ₘ = 0 := by
rw [← sub_eq_zero, ← LinearMap.map_sub, sub_mulVec]
lemma matrix_eq_iff_eq_forall' : (∀ v, Λ *ᵥ v= Λ' *ᵥ v) ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ := by
refine Iff.intro (fun h ↦ fun w v ↦ ?_) (fun h ↦ fun v ↦ ?_)
· rw [h w]
· simp only [matrix_apply_eq_iff_sub] at h
refine sub_eq_zero.1 ?_
have h1 := h v
rw [nondegenerate] at h1
simp only [sub_mulVec] at h1
exact h1
lemma matrix_eq_iff_eq_forall : Λ = Λ' ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ := by
rw [← matrix_eq_iff_eq_forall']
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact fun _ => congrFun (congrArg mulVec h) _
· rw [← (LinearMap.toMatrix stdBasis stdBasis).toEquiv.symm.apply_eq_iff_eq]
ext x
simp only [LinearEquiv.coe_toEquiv_symm, LinearMap.toMatrix_symm, EquivLike.coe_coe,
toLin_apply, h, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton]
lemma matrix_eq_id_iff : Λ = 1 ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, w⟫ₘ := by
rw [matrix_eq_iff_eq_forall]
simp
/-!
# The Minkowski metric and the standard basis
-/
| @[simp]
lemma basis_left (μ : Fin 1 ⊕ Fin d) : ⟪e μ, v⟫ₘ = η μ μ * v μ | minkowskiMetric.basis_left | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 315,
"tokenPositionInFile": 9906,
"theoremPositionInFile": 34
} | {
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"Lean.Elab.Deriving.Nonempty",
"Lean.Elab.Deriving.TypeName",
"Lean.Elab.Deriving.BEq",
"Lean.Meta.Inductive",
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"Lean.Elab.Deriving.Repr",
"Lean.Elab.Deriving.FromToJson",
"Lean.Elab.Deriving.SizeOf",
"Lean.Elab.Deriving.Hashable",
"Lean.Elab.Deriving.Ord",
"Lean.Elab.Deriving",
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"Lean.Elab.GenInjective",
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"Lean",
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"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
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"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
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"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
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"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
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"Std.Linter",
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"Std.Tactic.Classical",
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"Std.Tactic.SqueezeScope",
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"Std",
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"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
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"Mathlib.Algebra.BigOperators.List.Defs",
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"Mathlib.Tactic.Core",
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"Mathlib.Logic.Basic",
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"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
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"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
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"Aesop.Rule",
"Aesop.RuleSet.Member",
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"Lean.Replay",
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"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
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"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
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"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
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"Aesop.Stats.Extension",
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"Aesop.Frontend.Tactic",
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"Aesop",
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"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
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"Mathlib.Logic.Unique",
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"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
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"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
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"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
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"Qq.Macro",
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"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
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"Mathlib.Lean.Meta.Basic",
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"Mathlib.Tactic.ByContra",
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"Mathlib.Tactic.Choose",
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"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
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"Mathlib.Tactic.Existsi",
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"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
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"Mathlib.Algebra.Group.Units",
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"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
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"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
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"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
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"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
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"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
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"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
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"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Tactic.Peel",
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"Mathlib.GroupTheory.Perm.Support",
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"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
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"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
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"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
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"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
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"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
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"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
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"Mathlib.LinearAlgebra.Multilinear.Basis",
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"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
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"Mathlib.Topology.Sequences",
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"Mathlib.Topology.MetricSpace.DilationEquiv",
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"Mathlib.Analysis.NormedSpace.Basic",
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"Mathlib.Algebra.Star.Center",
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"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
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"Mathlib.Order.Copy",
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"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
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"Mathlib.Topology.GDelta",
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"Mathlib.Analysis.Convex.Slope",
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"Mathlib.LinearAlgebra.SymplecticGroup",
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]
} | {
"hasProof": true,
"proof": ":= by\n rw [as_sum]\n rcases μ with μ | μ\n · fin_cases μ\n simp [stdBasis_apply, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]\n · simp [stdBasis_apply, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]",
"proofType": "tactic",
"proofLengthLines": 5,
"proofLengthTokens": 227
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space time
This file introduce 4d Minkowski spacetime.
-/
noncomputable section
/-- The space-time -/
def SpaceTime : Type := Fin 4 → ℝ
/-- Give spacetime the structure of an additive commutative monoid. -/
instance : AddCommMonoid SpaceTime := Pi.addCommMonoid
/-- Give spacetime the structure of a module over the reals. -/
instance : Module ℝ SpaceTime := Pi.module _ _ _
instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normedAddCommGroup
instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
namespace SpaceTime
open Manifold
open Matrix
open Complex
open ComplexConjugate
/-- The space part of spacetime. -/
@[simp]
def space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
/-- The structure of a smooth manifold on spacetime. -/
def asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime := 𝓘(ℝ, SpaceTime)
instance : ChartedSpace SpaceTime SpaceTime := chartedSpaceSelf SpaceTime
/-- The standard basis for spacetime. -/
def stdBasis : Basis (Fin 4) ℝ SpaceTime := Pi.basisFun ℝ (Fin 4)
lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 := by
erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']
lemma stdBasis_not_eq {μ ν : Fin 4} (h : μ ≠ ν) : stdBasis μ ν = 0 := by
rw [stdBasis_apply]
exact if_neg h
lemma stdBasis_0 : stdBasis 0 = ![1, 0, 0, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_1 : stdBasis 1 = ![0, 1, 0, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_2 : stdBasis 2 = ![0, 0, 1, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_3 : stdBasis 3 = ![0, 0, 0, 1] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_mulVec (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
(Λ *ᵥ stdBasis μ) ν = Λ ν μ := by
rw [mulVec, dotProduct, Fintype.sum_eq_single μ, stdBasis_apply]
· simp only [↓reduceIte, mul_one]
· intro x h
rw [stdBasis_apply, if_neg (Ne.symm h)]
exact CommMonoidWithZero.mul_zero (Λ ν x)
| lemma explicit (x : SpaceTime) : x = ![x 0, x 1, x 2, x 3] | SpaceTime.explicit | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "b9119dccf1cb5708c9237596fe29562ca085614a",
"date": "2024-04-18T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
"lineInFile": 80,
"tokenPositionInFile": 2375,
"theoremPositionInFile": 11
} | {
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"Std.Lean.MonadBacktrack",
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"Lean.Replay",
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"Mathlib.Tactic.ExtractLets",
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"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
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"Mathlib.Tactic.Lift",
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"Mathlib.Lean.Name",
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"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
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"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
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"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
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"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
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"Mathlib.Data.Bracket",
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"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
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"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
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"Mathlib.Order.Max",
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"Mathlib.Order.ULift",
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"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.Order.LiminfLimsup",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Curry",
"Mathlib.Analysis.Calculus.FormalMultilinearSeries",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Analysis.Analytic.Basic",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Analysis.Analytic.Composition",
"Mathlib.Analysis.Analytic.Linear",
"Mathlib.Analysis.Analytic.CPolynomial",
"Mathlib.Analysis.Calculus.TangentCone",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics",
"Mathlib.Analysis.Calculus.FDeriv.Basic",
"Mathlib.Analysis.Calculus.Deriv.Basic",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.Calculus.FDeriv.Linear",
"Mathlib.Analysis.Calculus.FDeriv.Comp",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.ContDiff.Defs",
"Mathlib.Analysis.Calculus.FDeriv.Analytic",
"Mathlib.Geometry.Manifold.ChartedSpace",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.FDeriv.Bilinear",
"Mathlib.Analysis.Calculus.FDeriv.Mul",
"Mathlib.Analysis.Calculus.FDeriv.RestrictScalars",
"Mathlib.Analysis.Calculus.Deriv.Comp",
"Mathlib.Analysis.Calculus.Deriv.Inverse",
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Geometry.Manifold.SmoothManifoldWithCorners",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.FieldTheory.Tower",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2"
]
} | {
"hasProof": true,
"proof": ":= by\n funext i\n fin_cases i <;> rfl",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 38
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import HepLean.SpaceTime.LorentzGroup.Basic
import Mathlib.RepresentationTheory.Basic
/-!
# Covariant Lorentz vectors
The type `LorentzVector` corresponds to contravariant Lorentz tensors.
In this section we define covariant Lorentz tensors.
-/
/-! TODO: Define equivariant map between contravariant and covariant lorentz tensors. -/
noncomputable section
open BigOperators
/- The number of space dimensions . -/
variable (d : ℕ)
/-- The type of covariant Lorentz Vectors in `d`-space dimensions. -/
def CovariantLorentzVector : Type := (Fin 1 ⊕ Fin d) → ℝ
/-- An instance of an additive commutative monoid on `LorentzVector`. -/
instance : AddCommMonoid (CovariantLorentzVector d) := Pi.addCommMonoid
/-- An instance of a module on `LorentzVector`. -/
noncomputable instance instModuleRealCovariantLorentzVector : Module ℝ (CovariantLorentzVector d) := Pi.module _ _ _
instance : AddCommGroup (CovariantLorentzVector d) := Pi.addCommGroup
/-- The structure of a topological space `LorentzVector d`. -/
instance : TopologicalSpace (CovariantLorentzVector d) :=
haveI : NormedAddCommGroup (CovariantLorentzVector d) := Pi.normedAddCommGroup
UniformSpace.toTopologicalSpace
namespace CovariantLorentzVector
variable {d : ℕ} (v : CovariantLorentzVector d)
/-- The standard basis of `LorentzVector` indexed by `Fin 1 ⊕ Fin (d)`. -/
@[simps!]
noncomputable def stdBasis : Basis (Fin 1 ⊕ Fin (d)) ℝ (CovariantLorentzVector d) := Pi.basisFun ℝ _
lemma decomp_stdBasis (v : CovariantLorentzVector d) : ∑ i, v i • stdBasis i = v := by
funext ν
rw [Finset.sum_apply, Finset.sum_eq_single_of_mem ν]
· simp only [HSMul.hSMul, SMul.smul, stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ ν
· intros b _ hbi
simp only [HSMul.hSMul, SMul.smul, stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [Pi.single]
apply Or.inr (Function.update_noteq (id (Ne.symm hbi)) 1 0)
@[simp]
lemma decomp_stdBasis' (v : CovariantLorentzVector d) :
v (Sum.inl 0) • stdBasis (Sum.inl 0)
+ ∑ a₂ : Fin d, v (Sum.inr a₂) • stdBasis (Sum.inr a₂) = v := by
trans ∑ i, v i • stdBasis i
· simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,
Finset.sum_singleton]
· exact decomp_stdBasis v
/-!
## Lorentz group action on covariant Lorentz vectors
-/
/-- The representation of the Lorentz group acting on covariant Lorentz vectors. -/
def rep : Representation ℝ (LorentzGroup d) (CovariantLorentzVector d) where
toFun g := Matrix.toLinAlgEquiv stdBasis (LorentzGroup.transpose g⁻¹)
map_one' := by
simp only [inv_one, LorentzGroup.transpose_one, lorentzGroupIsGroup_one_coe, map_one]
map_mul' x y := by
simp only [mul_inv_rev, lorentzGroupIsGroup_inv, LorentzGroup.transpose_mul,
lorentzGroupIsGroup_mul_coe, map_mul]
open Matrix in
lemma rep_apply (g : LorentzGroup d) : rep g v = (g.1⁻¹)ᵀ *ᵥ v := by
trans (LorentzGroup.transpose g⁻¹) *ᵥ v
· rfl
· apply congrFun
apply congrArg
simp only [LorentzGroup.transpose, lorentzGroupIsGroup_inv, transpose_inj]
rw [← LorentzGroup.coe_inv]
rfl
| lemma rep_apply_stdBasis (g : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :
rep g (stdBasis μ) = ∑ ν, g⁻¹.1 μ ν • stdBasis ν | CovariantLorentzVector.rep_apply_stdBasis | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Covariant.lean | HepLean.SpaceTime.LorentzVector.Covariant | HepLean.SpaceTime.LorentzVector.Covariant.jsonl | {
"lineInFile": 98,
"tokenPositionInFile": 3379,
"theoremPositionInFile": 5
} | {
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"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
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"Mathlib.GroupTheory.GroupAction.BigOperators",
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"Mathlib.Data.Rat.BigOperators",
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"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
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"Mathlib.Algebra.Order.Nonneg.Field",
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"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
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"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
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"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
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"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
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"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
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"Mathlib.Order.LiminfLimsup",
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"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
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"Mathlib.Topology.Order.T5",
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"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
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"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
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"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
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"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
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"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
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"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
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"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
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"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
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"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
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"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
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"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
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"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
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"Mathlib.Data.Polynomial.Identities",
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"Mathlib.FieldTheory.Minpoly.Field",
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"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
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"Mathlib.Algebra.Star.Center",
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"Mathlib.Algebra.Star.Subalgebra",
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"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
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"Mathlib.Topology.Instances.RealVectorSpace",
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"Mathlib.Data.Sign",
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"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
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"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
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"Mathlib.Analysis.Convex.Strict",
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"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
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"Mathlib.Analysis.Normed.Group.AddTorsor",
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"Mathlib.Analysis.NormedSpace.Pointwise",
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"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
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"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
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"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
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"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
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"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
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"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
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"Mathlib.Algebra.Lie.OfAssociative",
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"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
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"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric",
"HepLean.SpaceTime.LorentzVector.NormOne",
"HepLean.SpaceTime.LorentzGroup.Basic",
"Mathlib.LinearAlgebra.Dual",
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.RepresentationTheory.Basic"
]
} | {
"hasProof": true,
"proof": ":= by\n simp only [rep_apply, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,\n Finset.sum_singleton, decomp_stdBasis']\n funext ν\n simp only [lorentzGroupIsGroup_inv]\n erw [Pi.basisFun_apply, Matrix.mulVec_single_one]\n rw [← LorentzGroup.coe_inv]\n rfl",
"proofType": "tactic",
"proofLengthLines": 7,
"proofLengthTokens": 286
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space time
This file introduce 4d Minkowski spacetime.
-/
noncomputable section
/-- The space-time -/
def SpaceTime : Type := Fin 4 → ℝ
/-- Give spacetime the structure of an additive commutative monoid. -/
instance : AddCommMonoid SpaceTime := Pi.addCommMonoid
/-- Give spacetime the structure of a module over the reals. -/
instance : Module ℝ SpaceTime := Pi.module _ _ _
instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normedAddCommGroup
instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
namespace SpaceTime
open Manifold
open Matrix
open Complex
open ComplexConjugate
/-- The space part of spacetime. -/
@[simp]
def space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
/-- The structure of a smooth manifold on spacetime. -/
def asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime := 𝓘(ℝ, SpaceTime)
instance : ChartedSpace SpaceTime SpaceTime := chartedSpaceSelf SpaceTime
/-- The standard basis for spacetime. -/
def stdBasis : Basis (Fin 4) ℝ SpaceTime := Pi.basisFun ℝ (Fin 4)
lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 := by
erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']
| lemma stdBasis_not_eq {μ ν : Fin 4} (h : μ ≠ ν) : stdBasis μ ν = 0 | SpaceTime.stdBasis_not_eq | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "db0f6de59c636e635219f2013bf9cad19a2d09ca",
"date": "2024-05-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
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"Mathlib.Lean.Name",
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"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
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"Mathlib.Tactic.UnsetOption",
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"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
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"ProofWidgets.Data.Html",
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"ProofWidgets.Component.OfRpcMethod",
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"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
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"Mathlib.Data.Sum.Basic",
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"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
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"Mathlib.Data.Multiset.Basic",
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"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.Order.LiminfLimsup",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Curry",
"Mathlib.Analysis.Calculus.FormalMultilinearSeries",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Analysis.Analytic.Basic",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Analysis.Analytic.Composition",
"Mathlib.Analysis.Analytic.Linear",
"Mathlib.Analysis.Analytic.CPolynomial",
"Mathlib.Analysis.Calculus.TangentCone",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics",
"Mathlib.Analysis.Calculus.FDeriv.Basic",
"Mathlib.Analysis.Calculus.Deriv.Basic",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.Calculus.FDeriv.Linear",
"Mathlib.Analysis.Calculus.FDeriv.Comp",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.ContDiff.Defs",
"Mathlib.Analysis.Calculus.FDeriv.Analytic",
"Mathlib.Geometry.Manifold.ChartedSpace",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
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"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.FDeriv.Bilinear",
"Mathlib.Analysis.Calculus.FDeriv.Mul",
"Mathlib.Analysis.Calculus.FDeriv.RestrictScalars",
"Mathlib.Analysis.Calculus.Deriv.Comp",
"Mathlib.Analysis.Calculus.Deriv.Inverse",
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Geometry.Manifold.SmoothManifoldWithCorners",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.FieldTheory.Tower",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.RCLike.Lemmas",
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"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
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"Mathlib.Algebra.Order.ToIntervalMod",
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"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2"
]
} | {
"hasProof": true,
"proof": ":= by\n rw [stdBasis_apply]\n exact if_neg h",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 44
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.Algebra.Lie.Classical
/-!
# The Lorentz Algebra
We define
- Define `lorentzAlgebra` via `LieAlgebra.Orthogonal.so'` as a subalgebra of
`Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`.
- In `mem_iff` prove that a matrix is in the Lorentz algebra if and only if it satisfies the
condition `Aᵀ * η = - η * A`.
-/
namespace SpaceTime
open Matrix
open TensorProduct
/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`. -/
def lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :=
(LieAlgebra.Orthogonal.so' (Fin 1) (Fin 3) ℝ)
namespace lorentzAlgebra
open minkowskiMatrix
lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
have h1 := A.2
erw [mem_skewAdjointMatricesLieSubalgebra] at h1
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h1
| lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ}
(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra | SpaceTime.lorentzAlgebra.mem_of_transpose_eta_eq_eta_mul_self | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzAlgebra/Basic.lean | HepLean.SpaceTime.LorentzAlgebra.Basic | HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl | {
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"Lean.Elab.Tactic.Conv.Pattern",
"Lean.Elab.Tactic.Delta",
"Lean.Elab.Tactic.Conv.Delta",
"Lean.Meta.Tactic.Unfold",
"Lean.Elab.Tactic.Unfold",
"Lean.Elab.Tactic.Conv.Unfold",
"Lean.Elab.Tactic.Conv",
"Lean.Elab.Tactic.Meta",
"Lean.Elab.Tactic.Cache",
"Lean.Elab.Calc",
"Lean.Elab.Tactic.Calc",
"Lean.Elab.Tactic.Congr",
"Lean.Elab.Tactic.Guard",
"Lean.Elab.Tactic.Change",
"Lean.Elab.Tactic.FalseOrByContra",
"Lean.Elab.Tactic.Omega.OmegaM",
"Lean.Elab.Tactic.Omega.MinNatAbs",
"Lean.Elab.Tactic.Omega.Core",
"Lean.Elab.Tactic.Omega.Frontend",
"Lean.Elab.Tactic.Omega",
"Lean.Elab.Tactic.Simpa",
"Lean.Meta.Tactic.NormCast",
"Lean.Elab.Tactic.NormCast",
"Lean.Elab.Tactic.Symm",
"Lean.LabelAttribute",
"Lean.Meta.Iterator",
"Lean.Meta.Tactic.IndependentOf",
"Lean.Meta.Tactic.Backtrack",
"Lean.Meta.Tactic.SolveByElim",
"Lean.Elab.Tactic.SolveByElim",
"Lean.Meta.LazyDiscrTree",
"Lean.Util.Heartbeats",
"Lean.Meta.Tactic.LibrarySearch",
"Lean.Elab.Tactic.LibrarySearch",
"Lean.Elab.Tactic.ShowTerm",
"Lean.Elab.Tactic",
"Lean.Elab.StructInst",
"Lean.Elab.Print",
"Lean.Elab.PreDefinition.WF",
"Lean.Elab.PreDefinition",
"Lean.Elab.Deriving.Util",
"Lean.Elab.Deriving.Inhabited",
"Lean.Elab.Deriving.Nonempty",
"Lean.Elab.Deriving.TypeName",
"Lean.Elab.Deriving.BEq",
"Lean.Meta.Inductive",
"Lean.Elab.Deriving.DecEq",
"Lean.Elab.Deriving.Repr",
"Lean.Elab.Deriving.FromToJson",
"Lean.Elab.Deriving.SizeOf",
"Lean.Elab.Deriving.Hashable",
"Lean.Elab.Deriving.Ord",
"Lean.Elab.Deriving",
"Lean.Elab.Extra",
"Lean.Elab.GenInjective",
"Lean.Elab.Macro",
"Lean.Elab.Notation",
"Lean.Elab.Mixfix",
"Lean.Elab.MacroRules",
"Lean.Elab.BuiltinCommand",
"Lean.Elab.InheritDoc",
"Lean.Elab.ParseImportsFast",
"Lean.Elab.GuardMsgs",
"Lean.Elab.CheckTactic",
"Lean.Elab.MatchExpr",
"Lean.Elab",
"Lean.Meta.LevelDefEq",
"Lean.Meta.UnificationHint",
"Lean.Meta.ExprDefEq",
"Lean.Meta.Tactic.LinearArith.Solver",
"Lean.Meta.Tactic.LinearArith.Nat.Solver",
"Lean.Meta.Tactic.LinearArith.Nat",
"Lean.Meta.Tactic.LinearArith.Main",
"Lean.Meta.Tactic.LinearArith",
"Lean.Meta.Tactic.AC.Main",
"Lean.Meta.Tactic.AC",
"Lean.Meta.Tactic",
"Lean.Meta.ExprLens",
"Lean.Meta.ExprTraverse",
"Lean.Meta",
"Lean.Util.ShareCommon",
"Lean.Util.TestExtern",
"Lean.Util.LeanOptions",
"Lean.Util.FileSetupInfo",
"Lean.Util",
"Lean.Server.Watchdog",
"Lean.LoadDynlib",
"Lean.Server.FileWorker.WidgetRequests",
"Lean.Util.LakePath",
"Lean.Server.FileWorker.SetupFile",
"Lean.Server.ImportCompletion",
"Lean.Server.FileWorker",
"Lean.Server.Rpc.Deriving",
"Lean.Server.Rpc",
"Lean.Server",
"Lean.Widget",
"Lean.Linter.Builtin",
"Lean.Linter",
"Lean",
"Std.Classes.BEq",
"Std.Classes.Cast",
"Std.Classes.Order",
"Std.Classes.RatCast",
"Std.Classes.SatisfiesM",
"Std.CodeAction.Misc",
"Std.CodeAction",
"Std.Control.ForInStep.Basic",
"Std.Control.ForInStep.Lemmas",
"Std.Control.ForInStep",
"Std.Control.Lemmas",
"Std.Data.MLList.Basic",
"Std.Control.Nondet.Basic",
"Std.Data.List.Init.Attach",
"Std.Data.Array.Basic",
"Std.Data.Bool",
"Std.Data.Fin.Basic",
"Std.Data.Option.Lemmas",
"Std.Data.List.Lemmas",
"Std.Tactic.SeqFocus",
"Std.Util.ProofWanted",
"Std.Data.Array.Lemmas",
"Std.Data.Array.Merge",
"Std.Data.Array.Monadic",
"Std.Data.Array",
"Std.Data.AssocList",
"Std.Data.BinomialHeap.Basic",
"Std.Data.BinomialHeap.Lemmas",
"Std.Data.BinomialHeap",
"Std.Data.Fin.Lemmas",
"Std.Data.BitVec.Lemmas",
"Std.Data.BitVec",
"Std.Data.ByteArray",
"Std.Data.Char",
"Std.Data.DList",
"Std.Data.Fin",
"Std.Data.HashMap.Basic",
"Std.Data.HashMap.Lemmas",
"Std.Data.HashMap.WF",
"Std.Data.HashMap",
"Std.Data.Int.Gcd",
"Std.Data.Int.Lemmas",
"Std.Data.Int",
"Std.Data.LazyList",
"Std.Data.List.Count",
"Std.Data.List.Pairwise",
"Std.Data.List.Perm",
"Std.Data.List",
"Std.Data.MLList.Heartbeats",
"Std.Lean.System.IO",
"Std.Data.MLList.IO",
"Std.Data.MLList",
"Std.Data.Nat",
"Std.Data.Option",
"Std.Data.PairingHeap",
"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric"
]
} | {
"hasProof": true,
"proof": ":= by\n erw [mem_skewAdjointMatricesLieSubalgebra]\n simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 124
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
@[simp]
lemma dual_id : @dual d 1 = 1 := by
simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
@[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
simp only [dual, transpose_mul]
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
· noncomm_ring [minkowskiMatrix.sq]
· noncomm_ring
@[simp]
lemma dual_dual : dual (dual Λ) = Λ := by
simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
trans (η * η) * Λ * (η * η)
· noncomm_ring
· noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_eta : @dual d η = η := by
simp only [dual, eq_transpose]
noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
noncomm_ring
@[simp]
lemma det_dual : (dual Λ).det = Λ.det := by
simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
group
norm_cast
simp
@[simp]
lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,
mulVec_mulVec, ← mul_assoc, minkowskiMatrix.sq, one_mul, (vecMul_transpose Λ x).symm, ←
dotProduct_mulVec]
@[simp]
lemma dual_mulVec_left : ⟪(dual Λ) *ᵥ x, y⟫ₘ = ⟪x, Λ *ᵥ y⟫ₘ := by
rw [symm, dual_mulVec_right, symm]
lemma matrix_apply_eq_iff_sub : ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ ↔ ⟪v, (Λ - Λ') *ᵥ w⟫ₘ = 0 := by
rw [← sub_eq_zero, ← LinearMap.map_sub, sub_mulVec]
| lemma matrix_eq_iff_eq_forall' : (∀ v, Λ *ᵥ v= Λ' *ᵥ v) ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ | minkowskiMetric.matrix_eq_iff_eq_forall' | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 285,
"tokenPositionInFile": 8879,
"theoremPositionInFile": 31
} | {
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"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
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"Mathlib.Data.Nat.PartENat",
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"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
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"Mathlib.Data.Nat.Prime",
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"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
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"Mathlib.Data.Sigma.Lex",
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"Mathlib.GroupTheory.Subsemigroup.Membership",
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"Mathlib.RingTheory.Subsemiring.Basic",
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"Mathlib.Order.Filter.Extr",
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"Mathlib.Algebra.Module.Submodule.LinearMap",
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"Mathlib.Data.Finset.NatAntidiagonal",
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"Mathlib.Data.Int.Parity",
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"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
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"Mathlib.Data.Fin.Interval",
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"Mathlib.Data.List.FinRange",
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"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
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"Mathlib.Algebra.Module.Hom",
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"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
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"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
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"Mathlib.Algebra.Module.Submodule.Pointwise",
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"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
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"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
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"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
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"Mathlib.Data.Int.ConditionallyCompleteOrder",
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"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
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"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
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"Mathlib.Data.ENNReal.Inv",
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"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
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"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
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"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
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"Mathlib.Topology.Order.MonotoneContinuity",
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"Mathlib.Data.Complex.Abs",
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"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
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"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
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"Mathlib.GroupTheory.Submonoid.Order",
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"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
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"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
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"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
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"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
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"Mathlib.LinearAlgebra.Isomorphisms",
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"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
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"Mathlib.RingTheory.Int.Basic",
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"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
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"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
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"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
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"Mathlib.GroupTheory.Perm.Closure",
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"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
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"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
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"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
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"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
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"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
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"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
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"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
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"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
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"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
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"Mathlib.Analysis.Convex.Combination",
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"Mathlib.Topology.Order.ProjIcc",
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"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
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"Mathlib.Topology.Algebra.FilterBasis",
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} | {
"hasProof": true,
"proof": ":= by\n refine Iff.intro (fun h ↦ fun w v ↦ ?_) (fun h ↦ fun v ↦ ?_)\n · rw [h w]\n · simp only [matrix_apply_eq_iff_sub] at h\n refine sub_eq_zero.1 ?_\n have h1 := h v\n rw [nondegenerate] at h1\n simp only [sub_mulVec] at h1\n exact h1",
"proofType": "tactic",
"proofLengthLines": 8,
"proofLengthTokens": 248
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space time
This file introduce 4d Minkowski spacetime.
-/
noncomputable section
/-- The space-time -/
def SpaceTime : Type := Fin 4 → ℝ
/-- Give spacetime the structure of an additive commutative monoid. -/
instance : AddCommMonoid SpaceTime := Pi.addCommMonoid
/-- Give spacetime the structure of a module over the reals. -/
instance : Module ℝ SpaceTime := Pi.module _ _ _
instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normedAddCommGroup
instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
namespace SpaceTime
open Manifold
open Matrix
open Complex
open ComplexConjugate
/-- The space part of spacetime. -/
@[simp]
def space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
/-- The structure of a smooth manifold on spacetime. -/
def asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime := 𝓘(ℝ, SpaceTime)
instance : ChartedSpace SpaceTime SpaceTime := chartedSpaceSelf SpaceTime
/-- The standard basis for spacetime. -/
def stdBasis : Basis (Fin 4) ℝ SpaceTime := Pi.basisFun ℝ (Fin 4)
| lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 | SpaceTime.stdBasis_apply | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "db0f6de59c636e635219f2013bf9cad19a2d09ca",
"date": "2024-05-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
"lineInFile": 49,
"tokenPositionInFile": 1386,
"theoremPositionInFile": 4
} | {
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"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
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"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
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"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
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"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
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"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
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"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
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"Mathlib.Topology.UniformSpace.UniformEmbedding",
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"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
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"Mathlib.GroupTheory.Subgroup.ZPowers",
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"Mathlib.Data.ZMod.Defs",
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"Mathlib.Data.Nat.SuccPred",
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"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
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"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
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"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
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"Mathlib.Data.Set.Semiring",
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"Mathlib.Algebra.Ring.OrderSynonym",
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"Mathlib.Order.Bounded",
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"Mathlib.SetTheory.Ordinal.Basic",
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"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
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"Mathlib.LinearAlgebra.Basis",
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"Mathlib.Data.ZMod.Basic",
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"Mathlib.Data.Sign",
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"Mathlib.Data.Finsupp.Fintype",
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"Mathlib.Data.Matrix.Basic",
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"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
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"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
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"Mathlib.Topology.Algebra.Affine",
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"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
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"Mathlib.Order.Filter.Subsingleton",
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"Mathlib.RingTheory.Finiteness",
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"Mathlib.RingTheory.AlgebraTower",
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"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
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"Mathlib.GroupTheory.NoncommPiCoprod",
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"Mathlib.Data.Nat.PrimeFin",
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"Mathlib.Data.Nat.MaxPowDiv",
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"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
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"Mathlib.Data.Fintype.Perm",
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"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
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"Mathlib.Init.Data.Prod",
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"Mathlib.RingTheory.Localization.FractionRing",
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"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
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"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
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"Mathlib.Data.W.Basic",
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"Mathlib.SetTheory.Cardinal.Subfield",
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"Mathlib.LinearAlgebra.Dimension.DivisionRing",
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"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
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"Mathlib.Data.Matrix.DMatrix",
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"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
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"Mathlib.Data.Matrix.Invertible",
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"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
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"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2"
]
} | {
"hasProof": true,
"proof": ":= by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']",
"proofType": "tactic",
"proofLengthLines": 1,
"proofLengthTokens": 68
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lorentz vectors
We define the contraction between a covariant and contravariant Lorentz vector,
as well as properties thereof.
The structures in this file are used in `HepLean.SpaceTime.LorentzTensor.Real.Basic`
to define the contraction between indices of Lorentz tensors.
-/
noncomputable section
open TensorProduct BigOperators
namespace LorentzVector
open Matrix
variable {d : ℕ} (v : LorentzVector d)
/-- The bi-linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
toFun v := {
toFun := fun w => ∑ i, v i * w i,
map_add' := by
intro w1 w2
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => mul_add _ _ _)
map_smul' := by
intro r w
simp only [RingHom.id_apply, smul_eq_mul]
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
ring}
map_add' v1 v2 := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl (fun i _ => add_mul _ _ _)
map_smul' r v := by
apply LinearMap.ext
intro w
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply]
rw [Finset.smul_sum]
refine Finset.sum_congr rfl (fun i _ => ?_)
simp only [HSMul.hSMul, SMul.smul]
exact mul_assoc r (v i) (w i)
/-- The linear map defining the contraction of a contravariant Lorentz vector
and a covariant Lorentz vector. -/
def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
TensorProduct.lift contrUpDownBi
lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rfl
@[simp]
lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· simp only [CovariantLorentzVector.stdBasis]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, mul_one]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
apply Or.inr hbi.symm
@[simp]
lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) :
contrUpDown (e i ⊗ₜ[ℝ] x) = x i := by
simp only [contrUpDown, contrUpDownBi, lift.tmul, LinearMap.coe_mk, AddHom.coe_mk]
rw [Finset.sum_eq_single_of_mem i]
· erw [Pi.basisFun_apply]
simp only [LinearMap.stdBasis_apply', ↓reduceIte, one_mul]
· exact Finset.mem_univ i
· intro b _ hbi
simp only [CovariantLorentzVector.stdBasis, mul_eq_zero]
erw [Pi.basisFun_apply]
left
simp only [LinearMap.stdBasis_apply', ite_eq_right_iff, one_ne_zero, imp_false]
exact hbi.symm
/-- The linear map defining the contraction of a covariant Lorentz vector
and a contravariant Lorentz vector. -/
def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
rw [dotProduct_comm x y]
rfl
/-!
## The unit of the contraction
-/
/-- The unit of the contraction of contravariant Lorentz vector and a
covariant Lorentz vector. -/
def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
simp only [unitUp, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
/-- The unit of the contraction of covariant Lorentz vector and a
contravariant Lorentz vector. -/
def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x := by
simp only [unitDown, LinearEquiv.refl_toLinearMap]
rw [tmul_sum]
simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,
lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']
/-!
# Contractions and the Lorentz actions
-/
open Matrix
| @[simp]
lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) | LorentzVector.contrUpDown_invariant_lorentzAction | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a65fb06605cbfe5b3907ae8a3963ce7cd664a975",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 152,
"tokenPositionInFile": 5616,
"theoremPositionInFile": 10
} | {
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"numInFilePremises": 2,
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"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
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"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
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"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
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"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
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"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
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"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
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"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
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"Mathlib.Data.Matrix.DMatrix",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
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"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
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"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
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"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
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"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
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"Mathlib.RingTheory.NonUnitalSubring.Basic",
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"Mathlib.Topology.MetricSpace.Dilation",
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"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
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} | {
"hasProof": true,
"proof": ":= by\n rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]\n simp only [rep_apply, CovariantLorentzVector.rep_apply]\n rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]\n simp only [LorentzGroup.subtype_inv_mul, one_mulVec]",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 253
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
| /-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i | minkowskiMetric.as_sum | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 135,
"tokenPositionInFile": 3921,
"theoremPositionInFile": 9
} | {
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"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
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} | {
"hasProof": true,
"proof": ":= by\n simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,\n dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,\n Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,\n Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,\n Function.comp_apply, minkowskiMatrix]\n rfl",
"proofType": "tactic",
"proofLengthLines": 6,
"proofLengthTokens": 416
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.
-/
/-! TODO: Prove topological properties of the Orthochronous Lorentz Group. -/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
variable {d : ℕ}
variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
lemma IsOrthochronous_iff_ge_one :
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
NormOneLorentzVector.time_pos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_neg_one :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ -1 := by
rw [timeComp, IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.not_mem_iff,
NormOneLorentzVector.time_nonpos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
rfl
lemma not_orthochronous_iff_le_zero :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [not_orthochronous_iff_le_neg_one] at h
linarith
· rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
def stepFunction : ℝ → ℝ := fun t =>
if t ≤ -1 then -1 else
if 1 ≤ t then 1 else t
| lemma stepFunction_continuous : Continuous stepFunction | LorentzGroup.stepFunction_continuous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "fdf5fda1e73e043ba40a3836ed1f00ddcc259373",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 74,
"tokenPositionInFile": 2499,
"theoremPositionInFile": 8
} | {
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"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
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"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
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"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
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"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
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"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
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"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
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"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
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"Mathlib.Analysis.LocallyConvex.Bounded",
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]
} | {
"hasProof": true,
"proof": ":= by\n apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)\n <;> intro a ha\n · rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha\n rw [ha]\n simp only [le_neg_self_iff, id_eq]\n have h1 : ¬ (1 : ℝ) ≤ 0 := by simp\n exact Eq.symm (if_neg h1)\n · rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha\n exact id (Eq.symm ha)",
"proofType": "tactic",
"proofLengthLines": 9,
"proofLengthTokens": 395
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`.
We define the give a series of lemmas related to the determinant of the Lorentz group.
-/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
namespace LorentzGroup
open minkowskiMetric
variable {d : ℕ}
/-- The determinant of a member of the Lorentz group is `1` or `-1`. -/
lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
refine mul_self_eq_one_iff.mp ?_
simpa only [det_mul, det_dual, det_one] using congrArg det ((mem_iff_self_mul_dual).mp Λ.2)
local notation "ℤ₂" => Multiplicative (ZMod 2)
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
instance : DiscreteTopology ℤ₂ := by
exact forall_open_iff_discrete.mp fun _ => trivial
instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
instance _root_.Finite.instDiscreteTopology [TopologicalSpace X] [T1Space X] [Finite X] : DiscreteTopology X :=
discreteTopology_iff_forall_isClosed.mpr (· |>.toFinite.isClosed)
/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
@[simps!]
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
toFun x := if x = ⟨1, Set.mem_insert_of_mem (-1) rfl⟩
then Additive.toMul 0 else Additive.toMul (1 : ZMod 2)
continuous_toFun := continuous_of_discreteTopology
/-- The continuous map taking a Lorentz matrix to its determinant. -/
def detContinuous : C(𝓛 d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨Λ.1.det, Or.symm (LorentzGroup.det_eq_one_or_neg_one _)⟩,
continuous_toFun := by
refine Continuous.subtype_mk ?_ _
exact Continuous.matrix_det $
Continuous.comp' (continuous_iff_le_induced.mpr fun U a => a) continuous_id'
}
@[simp]
lemma _root_.toMul_eq_one {α : Type*} [One α] {x : Additive α} :
Additive.toMul x = 1 ↔ x = 0 :=
Iff.rfl
lemma detContinuous_eq_one (Λ : LorentzGroup d) :
detContinuous Λ = Additive.toMul 0 ↔ Λ.1.det = 1 := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
Subtype.mk.injEq, ite_eq_left_iff, toMul_eq_one]
simp only [toMul_zero, ite_eq_left_iff, toMul_eq_one]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· by_contra hn
have h' := h hn
change (1 : Fin 2) = (0 : Fin 2) at h'
simp only [Fin.isValue, one_ne_zero] at h'
· intro h'
exact False.elim (h' h)
lemma detContinuous_eq_zero (Λ : LorentzGroup d) :
detContinuous Λ = Additive.toMul (1 : ZMod 2) ↔ Λ.1.det = - 1 := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
Subtype.mk.injEq, Nat.reduceAdd]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· by_contra hn
rw [if_pos] at h
· change (0 : Fin 2) = (1 : Fin 2) at h
simp only [Fin.isValue, zero_ne_one] at h
· cases' det_eq_one_or_neg_one Λ with h2 h2
· simp_all only [ite_true]
· simp_all only [not_true_eq_false]
· rw [if_neg]
· rfl
· cases' det_eq_one_or_neg_one Λ with h2 h2
· rw [h]
linarith
· linarith
lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
detContinuous Λ = detContinuous Λ' ↔ Λ.1.det = Λ'.1.det := by
cases' det_eq_one_or_neg_one Λ with h1 h1
· rw [h1, (detContinuous_eq_one Λ).mpr h1]
cases' det_eq_one_or_neg_one Λ' with h2 h2
· rw [h2, (detContinuous_eq_one Λ').mpr h2]
simp only [toMul_zero]
· rw [h2, (detContinuous_eq_zero Λ').mpr h2]
erw [Additive.toMul.apply_eq_iff_eq]
change (0 : Fin 2) = (1 : Fin 2) ↔ _
simp only [Fin.isValue, zero_ne_one, false_iff]
linarith
· rw [h1, (detContinuous_eq_zero Λ).mpr h1]
cases' det_eq_one_or_neg_one Λ' with h2 h2
· rw [h2, (detContinuous_eq_one Λ').mpr h2]
erw [Additive.toMul.apply_eq_iff_eq]
change (1 : Fin 2) = (0 : Fin 2) ↔ _
simp only [Fin.isValue, one_ne_zero, false_iff]
linarith
· rw [h2, (detContinuous_eq_zero Λ').mpr h2]
simp only [Nat.reduceAdd]
/-- The representation taking a Lorentz matrix to its determinant. -/
@[simps!]
def detRep : 𝓛 d →* ℤ₂ where
toFun Λ := detContinuous Λ
map_one' := by
simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
lorentzGroupIsGroup_one_coe, det_one, coeForℤ₂_apply, ↓reduceIte]
map_mul' Λ1 Λ2 := by
simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul, det_mul, toMul_zero,
mul_ite, mul_one, ite_mul, one_mul]
cases' det_eq_one_or_neg_one Λ1 with h1 h1
· rw [(detContinuous_eq_one Λ1).mpr h1]
cases' det_eq_one_or_neg_one Λ2 with h2 h2
· rw [(detContinuous_eq_one Λ2).mpr h2]
apply (detContinuous_eq_one _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_one]
· rw [(detContinuous_eq_zero Λ2).mpr h2]
apply (detContinuous_eq_zero _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one]
· rw [(detContinuous_eq_zero Λ1).mpr h1]
cases' det_eq_one_or_neg_one Λ2 with h2 h2
· rw [(detContinuous_eq_one Λ2).mpr h2]
apply (detContinuous_eq_zero _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_one]
· rw [(detContinuous_eq_zero Λ2).mpr h2]
apply (detContinuous_eq_one _).mpr
simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one, neg_neg]
| lemma detRep_continuous : Continuous (@detRep d) | LorentzGroup.detRep_continuous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
"lineInFile": 149,
"tokenPositionInFile": 5588,
"theoremPositionInFile": 6
} | {
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"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Proper
import Mathlib.Topology.Constructions
import HepLean.SpaceTime.LorentzVector.NormOne
/-!
# Boosts
This file defines those Lorentz which are boosts.
We first define generalised boosts, which are restricted lorentz transformations taking
a four velocity `u` to a four velocity `v`.
A boost is the special case of a generalised boost when `u = stdBasis 0`.
## References
- The main argument follows: Guillem Cobos, The Lorentz Group, 2015:
https://diposit.ub.edu/dspace/bitstream/2445/68763/2/memoria.pdf
-/
/-! TODO: Show that generalised boosts are in the restricted Lorentz group. -/
noncomputable section
namespace LorentzGroup
open NormOneLorentzVector
open minkowskiMetric
variable {d : ℕ}
/-- An auxillary linear map used in the definition of a generalised boost. -/
def genBoostAux₁ (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d where
toFun x := (2 * ⟪x, u⟫ₘ) • v.1.1
map_add' x y := by
simp only [map_add, LinearMap.add_apply]
rw [mul_add, add_smul]
map_smul' c x := by
simp only [map_smul, LinearMap.smul_apply, smul_eq_mul,
RingHom.id_apply]
rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
/-- An auxillary linear map used in the definition of a genearlised boost. -/
def genBoostAux₂ (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d where
toFun x := - (⟪x, u.1.1 + v⟫ₘ / (1 + ⟪u.1.1, v⟫ₘ)) • (u.1.1 + v)
map_add' x y := by
simp only
rw [← add_smul]
apply congrFun (congrArg _ _)
field_simp
apply congrFun (congrArg _ _)
ring
map_smul' c x := by
simp only
rw [map_smul, show (c • minkowskiMetric x) (↑u + ↑v) = c * minkowskiMetric x (u+v) from rfl,
mul_div_assoc, neg_mul_eq_mul_neg, smul_smul]
rfl
/-- An generalised boost. This is a Lorentz transformation which takes the four velocity `u`
to `v`. -/
def genBoost (u v : FuturePointing d) : LorentzVector d →ₗ[ℝ] LorentzVector d :=
LinearMap.id + genBoostAux₁ u v + genBoostAux₂ u v
namespace genBoost
/--
This lemma states that for a given four-velocity `u`, the general boost
transformation `genBoost u u` is equal to the identity linear map `LinearMap.id`.
-/
lemma self (u : FuturePointing d) : genBoost u u = LinearMap.id := by
ext x
simp only [genBoost, LinearMap.add_apply, LinearMap.id_coe, id_eq]
rw [add_assoc, add_right_eq_self, add_eq_zero_iff_eq_neg, genBoostAux₁, genBoostAux₂]
simp only [LinearMap.coe_mk, AddHom.coe_mk, map_add, smul_add, neg_smul, neg_add_rev, neg_neg]
rw [← add_smul]
apply congrFun (congrArg _ _)
rw [u.1.2]
ring
/-- A generalised boost as a matrix. -/
def toMatrix (u v : FuturePointing d) : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LinearMap.toMatrix LorentzVector.stdBasis LorentzVector.stdBasis (genBoost u v)
lemma toMatrix_mulVec (u v : FuturePointing d) (x : LorentzVector d) :
(toMatrix u v).mulVec x = genBoost u v x :=
LinearMap.toMatrix_mulVec_repr LorentzVector.stdBasis LorentzVector.stdBasis (genBoost u v) x
open minkowskiMatrix LorentzVector in
@[simp]
lemma toMatrix_apply (u v : FuturePointing d) (μ ν : Fin 1 ⊕ Fin d) :
(toMatrix u v) μ ν = η μ μ * (⟪e μ, e ν⟫ₘ + 2 * ⟪e ν, u⟫ₘ * ⟪e μ, v⟫ₘ
- ⟪e μ, u + v⟫ₘ * ⟪e ν, u + v⟫ₘ / (1 + ⟪u, v.1.1⟫ₘ)) := by
rw [matrix_apply_stdBasis (toMatrix u v) μ ν, toMatrix_mulVec]
simp only [genBoost, genBoostAux₁, genBoostAux₂, map_add, smul_add, neg_smul, LinearMap.add_apply,
LinearMap.id_apply, LinearMap.coe_mk, AddHom.coe_mk, basis_left, map_smul, smul_eq_mul, map_neg,
mul_eq_mul_left_iff]
ring_nf
exact (true_or_iff (η μ μ = 0)).mpr trivial
open minkowskiMatrix LorentzVector in
| lemma toMatrix_continuous (u : FuturePointing d) : Continuous (toMatrix u) | LorentzGroup.genBoost.toMatrix_continuous | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Boosts.lean | HepLean.SpaceTime.LorentzGroup.Boosts | HepLean.SpaceTime.LorentzGroup.Boosts.jsonl | {
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"Std.Data.PairingHeap",
"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
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} | {
"hasProof": true,
"proof": ":= by\n refine continuous_matrix ?_\n intro i j\n simp only [toMatrix_apply]\n refine (continuous_mul_left (η i i)).comp' ?_\n refine Continuous.sub ?_ ?_\n · refine .comp' (continuous_add_left (minkowskiMetric (e i) (e j))) ?_\n refine .comp' (continuous_mul_left (2 * ⟪e j, u⟫ₘ)) ?_\n exact FuturePointing.metric_continuous _\n refine .mul ?_ ?_\n · refine .mul ?_ ?_\n · simp only [(minkowskiMetric _).map_add]\n refine .comp' ?_ ?_\n · exact continuous_add_left ((minkowskiMetric (stdBasis i)) ↑u)\n · exact FuturePointing.metric_continuous _\n · simp only [(minkowskiMetric _).map_add]\n refine .comp' ?_ ?_\n · exact continuous_add_left _\n · exact FuturePointing.metric_continuous _\n · refine .inv₀ ?_ ?_\n · refine .comp' (continuous_add_left 1) (FuturePointing.metric_continuous _)\n exact fun x => FuturePointing.one_add_metric_non_zero u x",
"proofType": "tactic",
"proofLengthLines": 21,
"proofLengthTokens": 887
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
| @[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d | minkowskiMatrix.det_eq_neg_one_pow_d | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"tokenPositionInFile": 1776,
"theoremPositionInFile": 4
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"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
"Std.Data.Range",
"Std.Data.Rat.Lemmas",
"Std.Data.Rat",
"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
"Std.Data.Sum.Lemmas",
"Std.Data.Sum",
"Std.Data.UInt",
"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
"Std.Lean.TagAttribute",
"Std.Lean.AttributeExtra",
"Std.Lean.Delaborator",
"Std.Lean.Except",
"Std.Lean.Float",
"Std.Lean.HashMap",
"Std.Lean.HashSet",
"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
"Std.Lean.Meta.DiscrTree",
"Std.Lean.Meta.Inaccessible",
"Std.Lean.Meta.InstantiateMVars",
"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
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} | {
"hasProof": true,
"proof": ":= by\n simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]",
"proofType": "tactic",
"proofLengthLines": 1,
"proofLengthTokens": 72
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
| @[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d | minkowskiMatrix.eq_transpose | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 57,
"tokenPositionInFile": 1610,
"theoremPositionInFile": 3
} | {
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"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
"Std.Tactic.Basic",
"Std.Tactic.Case",
"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
"Std.Tactic.PrintDependents",
"Std.Tactic.PrintPrefix",
"Std.Tactic.SqueezeScope",
"Std.Tactic.Where",
"Std.Test.Internal.DummyLabelAttr",
"Std.Util.Cache",
"Std.Util.CheckTactic",
"Std.Util.ExtendedBinder",
"Std.Util.Pickle",
"Std.WF",
"Std",
"Mathlib.Tactic.PPWithUniv",
"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
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} | {
"hasProof": true,
"proof": ":= by\n simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]",
"proofType": "tactic",
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} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space time
This file introduce 4d Minkowski spacetime.
-/
noncomputable section
/-- The space-time -/
def SpaceTime : Type := Fin 4 → ℝ
/-- Give spacetime the structure of an additive commutative monoid. -/
instance : AddCommMonoid SpaceTime := Pi.addCommMonoid
/-- Give spacetime the structure of a module over the reals. -/
instance : Module ℝ SpaceTime := Pi.module _ _ _
instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normedAddCommGroup
instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
namespace SpaceTime
open Manifold
open Matrix
open Complex
open ComplexConjugate
/-- The space part of spacetime. -/
@[simp]
def space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
/-- The structure of a smooth manifold on spacetime. -/
def asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime := 𝓘(ℝ, SpaceTime)
instance : ChartedSpace SpaceTime SpaceTime := chartedSpaceSelf SpaceTime
/-- The standard basis for spacetime. -/
def stdBasis : Basis (Fin 4) ℝ SpaceTime := Pi.basisFun ℝ (Fin 4)
lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 := by
erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']
lemma stdBasis_not_eq {μ ν : Fin 4} (h : μ ≠ ν) : stdBasis μ ν = 0 := by
rw [stdBasis_apply]
exact if_neg h
lemma stdBasis_0 : stdBasis 0 = ![1, 0, 0, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
lemma stdBasis_1 : stdBasis 1 = ![0, 1, 0, 0] := by
funext i
fin_cases i <;> simp [stdBasis_apply]
| lemma stdBasis_2 : stdBasis 2 = ![0, 0, 1, 0] | SpaceTime.stdBasis_2 | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "db0f6de59c636e635219f2013bf9cad19a2d09ca",
"date": "2024-05-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
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"tokenPositionInFile": 1852,
"theoremPositionInFile": 8
} | {
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"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
"Mathlib.Tactic.SplitIfs",
"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Data.Bool.Basic",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
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"Mathlib.Data.Real.Basic",
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"Mathlib.Algebra.Module.Basic",
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"Mathlib.GroupTheory.Subsemigroup.Basic",
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"Mathlib.GroupTheory.Subsemigroup.Center",
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"Mathlib.Order.Zorn",
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"Mathlib.Topology.Basic",
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"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
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"Mathlib.Data.Set.BoolIndicator",
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"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
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"Mathlib.Topology.Homeomorph",
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"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
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"Mathlib.Algebra.Module.Equiv",
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"Mathlib.Topology.Algebra.InfiniteSum.Basic",
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"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
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"Mathlib.Topology.UniformSpace.Cauchy",
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"Mathlib.Topology.UniformSpace.UniformEmbedding",
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"Mathlib.Topology.UniformSpace.Pi",
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"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
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"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
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"Mathlib.Data.List.NatAntidiagonal",
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"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
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"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
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"Mathlib.Data.Finsupp.Indicator",
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"Mathlib.Data.List.FinRange",
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"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
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"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
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"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
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"Mathlib.Data.DFinsupp.Basic",
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"Mathlib.Data.Complex.Exponential",
"Mathlib.Algebra.Algebra.Tower",
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"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
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"Mathlib.RingTheory.NonUnitalSubring.Basic",
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"Mathlib.Topology.Defs.Sequences",
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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| @[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 | minkowskiMatrix.η_apply_mul_η_apply_diag | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Mathlib.Tactic.Cases",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Init.Control.Combinators",
"Mathlib.Data.Option.Defs",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Relator",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
"Aesop.Tracing",
"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
"Aesop.Builder.Basic",
"Aesop.Builder.Apply",
"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
"Aesop.Builder.Constructors",
"Aesop.Builder.NormSimp",
"Aesop.Builder.Tactic",
"Aesop.Builder.Default",
"Aesop.Builder.Forward",
"Aesop.Builder.Unfold",
"Aesop.Builder",
"Aesop.Frontend.RuleExpr",
"Aesop.Frontend.Attribute",
"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
"Aesop.Constants",
"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Data.Option.Basic",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Control.Functor",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Tactic.Inhabit",
"Mathlib.Logic.Unique",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Data.Prod.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Tactic.Spread",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.SimpRw",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.Coe",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lift",
"Mathlib.Tactic.Lint",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Data.Sum.Basic",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Init.Data.Quot",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Algebra.Opposites",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Data.Option.NAry",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Order.BoundedOrder",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Order.Disjoint",
"Mathlib.Order.WithBot",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Int.Basic",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
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"Mathlib.Topology.Algebra.InfiniteSum.Basic",
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"Mathlib.Data.Nat.SuccPred",
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"Mathlib.Tactic.Group",
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"Mathlib.GroupTheory.Coset",
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"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
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"Mathlib.Data.Set.Intervals.Infinite",
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"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
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"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
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"Mathlib.Data.DFinsupp.Basic",
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"Mathlib.Data.Finsupp.Basic",
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"Mathlib.LinearAlgebra.Pi",
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"Mathlib.LinearAlgebra.AffineSpace.Basic",
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"Mathlib.GroupTheory.Submonoid.Order",
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"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
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"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
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"Mathlib.LinearAlgebra.StdBasis",
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"Mathlib.LinearAlgebra.FreeModule.Basic",
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"Mathlib.RingTheory.Congruence",
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"Mathlib.RingTheory.Nilpotent",
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"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical"
]
} | {
"hasProof": true,
"proof": ":= by\n match μ with\n | Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]\n | Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]",
"proofType": "tactic",
"proofLengthLines": 3,
"proofLengthTokens": 184
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on spacetime in the mainly-minus signature.
We define the Minkowski metric as a bilinear map on the vector space
of Lorentz vectors in d dimensions.
-/
open Matrix BigOperators
/-!
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ
namespace minkowskiMatrix
variable {d : ℕ}
/-- Notation for `minkowskiMatrix`. -/
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
· simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inl.injEq, not_false_eq_true, one_apply_ne]
· rfl
· rfl
· simp only [diagonal, of_apply, Sum.inr.injEq, Sum.elim_inr, mul_neg, mul_one, neg_neg]
split
· rename_i h
subst h
simp_all only [one_apply_eq]
· simp_all only [ne_eq, Sum.inr.injEq, not_false_eq_true, one_apply_ne]
@[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 := by
match μ with
| Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
| Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]
refine fromBlocks_inj.mpr ?_
simp only [diagonal_one, true_and]
funext i j
rw [← diagonal_neg]
rfl
@[simp]
lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
exact diagonal_apply_ne _ h
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
map_add' y z := by
noncomm_ring
rw [mulVec_add, dotProduct_add]
map_smul' c y := by
simp only [RingHom.id_apply, smul_eq_mul]
rw [mulVec_smul, dotProduct_smul]
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
simp only [minkowskiLinearForm_apply, LinearMap.add_apply]
apply add_dotProduct
map_smul' c y := by
ext1 x
simp only [minkowskiLinearForm_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
rw [smul_dotProduct]
rfl
namespace minkowskiMetric
open minkowskiMatrix
open LorentzVector
variable {d : ℕ}
variable (v w : LorentzVector d)
/-- Notation for `minkowskiMetric`. -/
scoped[minkowskiMetric] notation:102 "⟪" v "," w "⟫ₘ" => minkowskiMetric v w
/-!
# Equalitites involving the Minkowski metric
-/
/-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,
dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,
Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,
Finset.sum_singleton, Sum.elim_inr, neg_mul, mul_neg, Finset.sum_neg_distrib, time, space,
Function.comp_apply, minkowskiMatrix]
rfl
/-- The Minkowski metric expressed as a sum for a single vector. -/
lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
rfl
lemma self_eq_time_minus_norm : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
/-- The Minkowski metric is symmetric in its arguments.. -/
lemma symm : ⟪v, w⟫ₘ = ⟪w, v⟫ₘ := by
simp only [as_sum]
ac_rfl
lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 := by
rw [self_eq_time_minus_norm]
exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))
/-!
# Minkowski metric and space reflections
-/
lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time + ⟪v.space, w.space⟫_ℝ := by
rw [eq_time_minus_inner_prod, spaceReflection_space, spaceReflection_time]
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
have h2 : 0 ≤ ⟪v.space, v.space⟫_ℝ := real_inner_self_nonneg
have h3 : v.time * v.time = 0 := by linarith [mul_self_nonneg v.time]
have h4 : ⟪v.space, v.space⟫_ℝ = 0 := by linarith
simp only [time, Fin.isValue, mul_eq_zero, or_self] at h3
rw [@inner_self_eq_zero] at h4
funext i
rcases i with i | i
· fin_cases i
exact h3
· simpa using congrFun h4 i
· rw [h]
exact LinearMap.map_zero₂ minkowskiMetric (spaceReflection 0)
/-!
# Non degeneracy of the Minkowski metric
-/
/-- The metric tensor is non-degenerate. -/
lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact (self_spaceReflection_eq_zero_iff _).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalities involving the Minkowski metric
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]
exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ
lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w⟫ₘ := by
apply le_trans _ (ge_abs_inner_product v w)
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
-/
section matrices
variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
/-- The dual of a matrix with respect to the Minkowski metric. -/
def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
@[simp]
lemma dual_id : @dual d 1 = 1 := by
simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
@[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
simp only [dual, transpose_mul]
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
· noncomm_ring [minkowskiMatrix.sq]
· noncomm_ring
@[simp]
lemma dual_dual : dual (dual Λ) = Λ := by
simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
trans (η * η) * Λ * (η * η)
· noncomm_ring
· noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_eta : @dual d η = η := by
simp only [dual, eq_transpose]
noncomm_ring [minkowskiMatrix.sq]
@[simp]
lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
noncomm_ring
@[simp]
lemma det_dual : (dual Λ).det = Λ.det := by
simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
group
norm_cast
simp
@[simp]
lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,
mulVec_mulVec, ← mul_assoc, minkowskiMatrix.sq, one_mul, (vecMul_transpose Λ x).symm, ←
dotProduct_mulVec]
@[simp]
lemma dual_mulVec_left : ⟪(dual Λ) *ᵥ x, y⟫ₘ = ⟪x, Λ *ᵥ y⟫ₘ := by
rw [symm, dual_mulVec_right, symm]
lemma matrix_apply_eq_iff_sub : ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ ↔ ⟪v, (Λ - Λ') *ᵥ w⟫ₘ = 0 := by
rw [← sub_eq_zero, ← LinearMap.map_sub, sub_mulVec]
lemma matrix_eq_iff_eq_forall' : (∀ v, Λ *ᵥ v= Λ' *ᵥ v) ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ := by
refine Iff.intro (fun h ↦ fun w v ↦ ?_) (fun h ↦ fun v ↦ ?_)
· rw [h w]
· simp only [matrix_apply_eq_iff_sub] at h
refine sub_eq_zero.1 ?_
have h1 := h v
rw [nondegenerate] at h1
simp only [sub_mulVec] at h1
exact h1
lemma matrix_eq_iff_eq_forall : Λ = Λ' ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ := by
rw [← matrix_eq_iff_eq_forall']
refine Iff.intro (fun h => ?_) (fun h => ?_)
· exact fun _ => congrFun (congrArg mulVec h) _
· rw [← (LinearMap.toMatrix stdBasis stdBasis).toEquiv.symm.apply_eq_iff_eq]
ext x
simp only [LinearEquiv.coe_toEquiv_symm, LinearMap.toMatrix_symm, EquivLike.coe_coe,
toLin_apply, h, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
Finset.sum_singleton]
lemma matrix_eq_id_iff : Λ = 1 ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, w⟫ₘ := by
rw [matrix_eq_iff_eq_forall]
simp
/-!
# The Minkowski metric and the standard basis
-/
@[simp]
lemma basis_left (μ : Fin 1 ⊕ Fin d) : ⟪e μ, v⟫ₘ = η μ μ * v μ := by
rw [as_sum]
rcases μ with μ | μ
· fin_cases μ
simp [stdBasis_apply, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
· simp [stdBasis_apply, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma on_timeVec : ⟪timeVec, @timeVec d⟫ₘ = 1 := by
simp only [timeVec, Fin.isValue, basis_left, minkowskiMatrix,
LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_apply_eq, Sum.elim_inl, stdBasis_apply,
↓reduceIte, mul_one]
lemma on_basis_mulVec (μ ν : Fin 1 ⊕ Fin d) : ⟪e μ, Λ *ᵥ e ν⟫ₘ = η μ μ * Λ μ ν := by
simp [basis_left, mulVec, dotProduct, stdBasis_apply]
| lemma on_basis (μ ν : Fin 1 ⊕ Fin d) : ⟪e μ, e ν⟫ₘ = η μ ν | minkowskiMetric.on_basis | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 331,
"tokenPositionInFile": 10585,
"theoremPositionInFile": 37
} | {
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"Mathlib.Lean.Name",
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"Mathlib.Tactic.Rename",
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"Mathlib.Init.Core",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.Substs",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
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"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
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"Mathlib.Data.List.Forall2",
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"Mathlib.Data.Sum.Basic",
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"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
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"Mathlib.Data.Nat.Cast.Defs",
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"Mathlib.Data.List.ProdSigma",
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"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
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"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
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"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
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"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
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"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
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"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
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"Mathlib.GroupTheory.Archimedean",
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"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical"
]
} | {
"hasProof": true,
"proof": ":= by\n rw [basis_left, stdBasis_apply]\n by_cases h : μ = ν\n · simp [h]\n · simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_apply_eq,\n mul_ite, mul_one, mul_zero, ne_eq, h, not_false_eq_true, diagonal_apply_ne, ite_eq_right_iff]\n exact fun a => False.elim (h (id (Eq.symm a)))",
"proofType": "tactic",
"proofLengthLines": 6,
"proofLengthTokens": 315
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.Algebra.Lie.Classical
/-!
# The Lorentz Algebra
We define
- Define `lorentzAlgebra` via `LieAlgebra.Orthogonal.so'` as a subalgebra of
`Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`.
- In `mem_iff` prove that a matrix is in the Lorentz algebra if and only if it satisfies the
condition `Aᵀ * η = - η * A`.
-/
namespace SpaceTime
open Matrix
open TensorProduct
/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`. -/
def lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :=
(LieAlgebra.Orthogonal.so' (Fin 1) (Fin 3) ℝ)
namespace lorentzAlgebra
open minkowskiMatrix
lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
have h1 := A.2
erw [mem_skewAdjointMatricesLieSubalgebra] at h1
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h1
lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ}
(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra := by
erw [mem_skewAdjointMatricesLieSubalgebra]
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h
lemma mem_iff {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} :
A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A :=
Iff.intro (fun h => transpose_eta ⟨A, h⟩) (fun h => mem_of_transpose_eta_eq_eta_mul_self h)
lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
A ∈ lorentzAlgebra ↔ A = - η * Aᵀ * η := by
rw [mem_iff]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· trans -η * (Aᵀ * η)
· rw [h]
trans (η * η) * A
· rw [minkowskiMatrix.sq]
noncomm_ring
· noncomm_ring
· noncomm_ring
· nth_rewrite 2 [h]
trans (η * η) * Aᵀ * η
· rw [minkowskiMatrix.sq]
noncomm_ring
· noncomm_ring
| lemma diag_comp (Λ : lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : Λ.1 μ μ = 0 | SpaceTime.lorentzAlgebra.diag_comp | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | {
"commit": "28c9086f0df4cb6b566a6adbd391e5b6fcb0ee7f",
"date": "2024-06-12T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzAlgebra/Basic.lean | HepLean.SpaceTime.LorentzAlgebra.Basic | HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl | {
"lineInFile": 62,
"tokenPositionInFile": 1979,
"theoremPositionInFile": 5
} | {
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"Mathlib.Tactic.SudoSetOption",
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"ProofWidgets.Data.Html",
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"ProofWidgets.Component.OfRpcMethod",
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"Mathlib.Data.List.Rotate",
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"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.Algebra.Module.Prod",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.LinearAlgebra.AffineSpace.Basic",
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
"Mathlib.Analysis.Convex.Star",
"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Data.Complex.Order",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic",
"Mathlib.Data.Sign",
"Mathlib.LinearAlgebra.AffineSpace.Combination",
"Mathlib.LinearAlgebra.AffineSpace.Independent",
"Mathlib.LinearAlgebra.AffineSpace.Basis",
"Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional",
"Mathlib.Analysis.Convex.Between",
"Mathlib.Analysis.Convex.Hull",
"Mathlib.Analysis.Convex.Combination",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.Convex.Jensen",
"Mathlib.Analysis.Convex.Strict",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.CompactOpen",
"Mathlib.Data.Set.Intervals.Instances",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Algebra.Affine",
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Analysis.Normed.Group.Pointwise",
"Mathlib.Analysis.Normed.Group.AddTorsor",
"Mathlib.Analysis.NormedSpace.AddTorsor",
"Mathlib.Analysis.Convex.Normed",
"Mathlib.Algebra.Order.Group.TypeTags",
"Mathlib.Analysis.Normed.Order.Basic",
"Mathlib.Analysis.NormedSpace.Real",
"Mathlib.Analysis.NormedSpace.Pointwise",
"Mathlib.Analysis.NormedSpace.Ray",
"Mathlib.Analysis.Convex.StrictConvexSpace",
"Mathlib.Analysis.Convex.Uniform",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Analysis.Normed.Group.Completion",
"Mathlib.Topology.Bornology.Absorbs",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Seminorm",
"Mathlib.Algebra.Module.LinearMap.Pointwise",
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.MetricSpace.Equicontinuity",
"Mathlib.Topology.Algebra.Module.LocallyConvex",
"Mathlib.Analysis.LocallyConvex.WithSeminorms",
"Mathlib.Topology.Algebra.Module.StrongTopology",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Basic",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm",
"Mathlib.Analysis.NormedSpace.Span",
"Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.UniformRing",
"Mathlib.Analysis.NormedSpace.Completion",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic",
"Mathlib.Analysis.NormedSpace.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Module",
"Mathlib.Analysis.Normed.Group.InfiniteSum",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Order.Copy",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.PartialHomeomorph",
"Mathlib.Analysis.Asymptotics.Asymptotics",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Analysis.NormedSpace.Units",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness",
"Mathlib.Analysis.NormedSpace.OperatorNorm.Mul",
"Mathlib.Analysis.NormedSpace.BoundedLinearMaps",
"Mathlib.Analysis.InnerProductSpace.Basic",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.Analysis.InnerProductSpace.Orthogonal",
"Mathlib.Topology.GDelta",
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Baire.CompleteMetrizable",
"Mathlib.LinearAlgebra.AffineSpace.Restrict",
"Mathlib.Analysis.NormedSpace.AffineIsometry",
"Mathlib.Analysis.NormedSpace.Banach",
"Mathlib.Analysis.InnerProductSpace.Symmetric",
"Mathlib.Analysis.NormedSpace.RCLike",
"Mathlib.Analysis.Asymptotics.Theta",
"Mathlib.Analysis.Asymptotics.AsymptoticEquivalent",
"Mathlib.Analysis.Normed.Group.Lemmas",
"Mathlib.Analysis.NormedSpace.RieszLemma",
"Mathlib.Control.Bifunctor",
"Mathlib.Logic.Equiv.Functor",
"Mathlib.Order.JordanHolder",
"Mathlib.Order.CompactlyGenerated.Intervals",
"Mathlib.RingTheory.SimpleModule",
"Mathlib.Topology.Algebra.Module.Simple",
"Mathlib.Data.Matrix.Invertible",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis",
"Mathlib.LinearAlgebra.Determinant",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Module.FiniteDimension",
"Mathlib.Topology.Instances.Matrix",
"Mathlib.Analysis.NormedSpace.FiniteDimension",
"Mathlib.Analysis.RCLike.Lemmas",
"Mathlib.Algebra.DirectSum.Decomposition",
"Mathlib.Analysis.InnerProductSpace.Projection",
"Mathlib.Analysis.Convex.Slope",
"Mathlib.Analysis.SpecialFunctions.Exp",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
"Mathlib.Analysis.Normed.Group.Quotient",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Algebra.ModEq",
"Mathlib.Order.Circular",
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Topology.Instances.AddCircle",
"Mathlib.Analysis.Normed.Group.AddCircle",
"Mathlib.Algebra.CharZero.Quotient",
"Mathlib.Topology.Instances.Sign",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle",
"Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse",
"Mathlib.Analysis.SpecialFunctions.Complex.Arg",
"Mathlib.Analysis.SpecialFunctions.Log.Basic",
"Mathlib.Analysis.SpecialFunctions.Complex.Log",
"Mathlib.Analysis.SpecialFunctions.Pow.Complex",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"Mathlib.Analysis.Convex.SpecificFunctions.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Data.Real.ConjExponents",
"Mathlib.Analysis.MeanInequalities",
"Mathlib.Order.Atoms.Finite",
"Mathlib.Data.Fintype.Order",
"Mathlib.Analysis.NormedSpace.WithLp",
"Mathlib.Analysis.NormedSpace.PiLp",
"Mathlib.LinearAlgebra.UnitaryGroup",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"HepLean.SpaceTime.LorentzVector.Basic",
"Mathlib.Algebra.Lie.Basic",
"Mathlib.Algebra.Lie.Subalgebra",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Algebra.Divisibility.Prod",
"Mathlib.RingTheory.Nakayama",
"Mathlib.RingTheory.Artinian",
"Mathlib.Algebra.Lie.Submodule",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.Algebra.Lie.IdealOperations",
"Mathlib.Algebra.Lie.Abelian",
"Mathlib.LinearAlgebra.Matrix.Transvection",
"Mathlib.Algebra.CharP.Reduced",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.RootsOfUnity.Basic",
"Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.Integer",
"Mathlib.LinearAlgebra.Matrix.ToLinearEquiv",
"Mathlib.Algebra.Lie.Matrix",
"Mathlib.LinearAlgebra.Basis.Bilinear",
"Mathlib.LinearAlgebra.Matrix.SesquilinearForm",
"Mathlib.Tactic.NoncommRing",
"Mathlib.Algebra.Lie.SkewAdjoint",
"Mathlib.LinearAlgebra.SymplecticGroup",
"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric"
]
} | {
"hasProof": true,
"proof": ":= by\n have h := congrArg (fun M ↦ M μ μ) $ mem_iff.mp Λ.2\n simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,\n transpose_apply, diagonal_neg, diagonal_mul, neg_mul] at h\n rcases μ with μ | μ\n · simp only [Sum.elim_inl, mul_one, one_mul] at h\n exact eq_zero_of_neg_eq (id (Eq.symm h))\n · simp only [Sum.elim_inr, mul_neg, mul_one, neg_mul, one_mul, neg_neg] at h\n exact eq_zero_of_neg_eq h",
"proofType": "tactic",
"proofLengthLines": 8,
"proofLengthTokens": 436
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.Algebra.Lie.Classical
/-!
# The Lorentz Algebra
We define
- Define `lorentzAlgebra` via `LieAlgebra.Orthogonal.so'` as a subalgebra of
`Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`.
- In `mem_iff` prove that a matrix is in the Lorentz algebra if and only if it satisfies the
condition `Aᵀ * η = - η * A`.
-/
namespace SpaceTime
open Matrix
open TensorProduct
/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`. -/
def lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :=
(LieAlgebra.Orthogonal.so' (Fin 1) (Fin 3) ℝ)
namespace lorentzAlgebra
open minkowskiMatrix
lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
have h1 := A.2
erw [mem_skewAdjointMatricesLieSubalgebra] at h1
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h1
lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ}
(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra := by
erw [mem_skewAdjointMatricesLieSubalgebra]
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h
lemma mem_iff {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} :
A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A :=
Iff.intro (fun h => transpose_eta ⟨A, h⟩) (fun h => mem_of_transpose_eta_eq_eta_mul_self h)
lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
A ∈ lorentzAlgebra ↔ A = - η * Aᵀ * η := by
rw [mem_iff]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· trans -η * (Aᵀ * η)
· rw [h]
trans (η * η) * A
· rw [minkowskiMatrix.sq]
noncomm_ring
· noncomm_ring
· noncomm_ring
· nth_rewrite 2 [h]
trans (η * η) * Aᵀ * η
· rw [minkowskiMatrix.sq]
noncomm_ring
· noncomm_ring
lemma diag_comp (Λ : lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : Λ.1 μ μ = 0 := by
have h := congrArg (fun M ↦ M μ μ) $ mem_iff.mp Λ.2
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
transpose_apply, diagonal_neg, diagonal_mul, neg_mul] at h
rcases μ with μ | μ
· simp only [Sum.elim_inl, mul_one, one_mul] at h
exact eq_zero_of_neg_eq (id (Eq.symm h))
· simp only [Sum.elim_inr, mul_neg, mul_one, neg_mul, one_mul, neg_neg] at h
exact eq_zero_of_neg_eq h
| lemma time_comps (Λ : lorentzAlgebra) (i : Fin 3) :
Λ.1 (Sum.inr i) (Sum.inl 0) = Λ.1 (Sum.inl 0) (Sum.inr i) | SpaceTime.lorentzAlgebra.time_comps | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | {
"commit": "28c9086f0df4cb6b566a6adbd391e5b6fcb0ee7f",
"date": "2024-06-12T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzAlgebra/Basic.lean | HepLean.SpaceTime.LorentzAlgebra.Basic | HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl | {
"lineInFile": 72,
"tokenPositionInFile": 2488,
"theoremPositionInFile": 6
} | {
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"ProofWidgets.Component.Basic",
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"ProofWidgets.Data.Html",
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"ProofWidgets.Component.OfRpcMethod",
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"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Data.List.GetD",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Data.Rat.Init",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Defs",
"Mathlib.Data.Rat.Order",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Algebra.Parity",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.GeomSum",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Order",
"Mathlib.Order.Filter.Basic",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.Iterate",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Module.Pi",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
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"Mathlib.Data.Real.Sqrt",
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"Mathlib.LinearAlgebra.AffineSpace.Basic",
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"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.LinearAlgebra.AffineSpace.Midpoint",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.GroupTheory.Submonoid.Order",
"Mathlib.RingTheory.Subring.Units",
"Mathlib.LinearAlgebra.Ray",
"Mathlib.Analysis.Convex.Segment",
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"Mathlib.LinearAlgebra.AffineSpace.AffineSubspace",
"Mathlib.Analysis.Convex.Basic",
"Mathlib.Algebra.DirectSum.Basic",
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"Mathlib.Data.Polynomial.Degree.Lemmas",
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"Mathlib.Topology.UnitInterval",
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"Mathlib.Topology.GDelta",
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"Mathlib.Algebra.Lie.Classical",
"HepLean.SpaceTime.MinkowskiMetric"
]
} | {
"hasProof": true,
"proof": ":= by\n simpa only [Fin.isValue, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,\n transpose_apply, Sum.elim_inr, mul_neg, mul_one, diagonal_neg, diagonal_mul, Sum.elim_inl,\n neg_mul, one_mul, neg_inj] using congrArg (fun M ↦ M (Sum.inl 0) (Sum.inr i)) $ mem_iff.mp Λ.2",
"proofType": "tactic",
"proofLengthLines": 3,
"proofLengthTokens": 299
} |