module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 539,
"column": 4
} | {
"line": 539,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\nF : Type u_5\ninst✝¹¹ : NormedA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 529,
"column": 2
} | {
"line": 547,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\nF : Type u_5\ninst✝¹¹ : NormedA... | let aux := fun x ↦ b.repr (e ((T% s) x)).2 i
-- Since `e.baseSet` is open, this is sufficient.
suffices MDiffAt aux x by
apply this.congr_of_eventuallyEq
apply eventuallyEq_of_mem (s := e.baseSet) (by simp [e.open_baseSet.mem_nhds hxe])
intro y hy
simp [aux, e.localFrame_coeff_eq_coeff hy]
simp on... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 529,
"column": 2
} | {
"line": 547,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\nF : Type u_5\ninst✝¹¹ : NormedA... | let aux := fun x ↦ b.repr (e ((T% s) x)).2 i
-- Since `e.baseSet` is open, this is sufficient.
suffices MDiffAt aux x by
apply this.congr_of_eventuallyEq
apply eventuallyEq_of_mem (s := e.baseSet) (by simp [e.open_baseSet.mem_nhds hxe])
intro y hy
simp [aux, e.localFrame_coeff_eq_coeff hy]
simp on... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 36
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : CompleteSpace 𝕜\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Polygon.Basic | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : AddTorsor V P\ninst✝¹ : Nontrivial R\nm : ℕ\ninst✝ : NeZero m.succ\npoly : Polygon P m.succ\nh : HasNondegenerateVertices R poly\ni : Fin m.succ\n⊢ poly.vertices i ≠ poly.vertices ((finRota... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Polygon.Basic | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 22
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\nt : Triangle R P\nht : t.points = ![t.points 0, t.points 1, t.points 2]\n⊢ AffineIndependent R ![t.points 0, t.points 1, t.points 2]",
"usedConstants": [
"Eq.mpr",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 17
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁰ : TopologicalSpace M\ninst✝⁹ : ChartedSpace H M\nF : Type u_5\ninst✝⁸ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ClassEquation | {
"line": 37,
"column": 51
} | {
"line": 37,
"column": 62
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype (ConjClasses G)\ninst✝¹ : Fintype G\ninst✝ : (x : ConjClasses G) → Fintype ↑x.carrier\nthis : (x : ConjClasses G) × ↑x.carrier ≃ G\n⊢ ∑ x, x.carrier.toFinset.card = Fintype.card G",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ClassEquation | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype (ConjClasses G)\ninst✝¹ : Fintype G\ninst✝ : (x : ConjClasses G) → Fintype ↑x.carrier\n⊢ (x : ConjClasses G) × ↑x.carrier ≃ G",
"usedConstants": [
"Eq.mpr",
"ConjClasses.carrier_eq_preimage_mk",
"congrArg",
"ConjClasses.mk",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ClassEquation | {
"line": 72,
"column": 39
} | {
"line": 72,
"column": 50
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nval✝ : Fintype G\nx✝ : ConjClasses G\ng : G\nhg : (carrier (Quot.mk (⇑(IsConj.setoid G)) g)).Subsingleton\n⊢ ↑(carrier (Quot.mk (⇑(IsConj.setoid G)) g)).toFinset = ↑{g}",
"usedConstants": [
"Eq.mpr",
"Finset.coe_singleton",
"IsConj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nF : Type u_5\ninst✝⁹ : NormedAd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁰ : TopologicalSpace M\ninst✝⁹ : ChartedSpace H M\nF : Type u_5\ninst✝⁸ : NormedAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : ChartedSpace H M\nF : Type u_5\ninst✝¹⁰ : NormedA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Commutator.Finite | {
"line": 41,
"column": 10
} | {
"line": 41,
"column": 21
} | [
{
"pp": "case pos\nη : Type u_2\ninst✝¹ : Finite η\nGs : η → Type u_3\ninst✝ : (i : η) → Group (Gs i)\nH K : (i : η) → Subgroup (Gs i)\nhi : (i : η) → Gs i\nj : η\n_hj : j ∈ Set.univ\nx : Gs j\nhx : x ∈ ↑(H j)\n⊢ (MonoidHom.mulSingle Gs j) x ∈ comap (Pi.evalMonoidHom Gs j) (H j)",
"usedConstants": [
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Commutator.Finite | {
"line": 41,
"column": 10
} | {
"line": 41,
"column": 21
} | [
{
"pp": "case pos\nη : Type u_2\ninst✝¹ : Finite η\nGs : η → Type u_3\ninst✝ : (i : η) → Group (Gs i)\nH K : (i : η) → Subgroup (Gs i)\nhi : (i : η) → Gs i\nj : η\n_hj : j ∈ Set.univ\nx : Gs j\nhx : x ∈ ↑(K j)\n⊢ (MonoidHom.mulSingle Gs j) x ∈ comap (Pi.evalMonoidHom Gs j) (K j)",
"usedConstants": [
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 98,
"column": 13
} | {
"line": 98,
"column": 24
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsKleinFour G\nh : IsCyclic G\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 15
} | [
{
"pp": "case inl\nn : ℕ\nhn : 0 < n\n⊢ ¬↑n = 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"AddMonoidWithOne.toNatCast",
"instOfNatNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 119,
"column": 15
} | {
"line": 119,
"column": 26
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ x * y ∉ {x, y, 1}",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"CancelMonoid.toRightCancelMonoid",
"InvOneClass.toOne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 54
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsKleinFour G\nx y z : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\nhz : z ≠ 1\nhzx : z ≠ x\nhzy : z ≠ y\nx✝ : Fintype G := ⋯\n⊢ z ∉ {x, y, 1}",
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 229,
"column": 22
} | {
"line": 229,
"column": 33
} | [
{
"pp": "x✝¹ : 0 ≠ 1\nx✝ : 0 ≠ 2\nh' : IsMulCommutative (DihedralGroup 0)\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 15
} | [
{
"pp": "n : ℕ\nx✝¹ : n + 3 ≠ 1\nx✝ : n + 3 ≠ 2\nh' : IsMulCommutative (DihedralGroup (n + 3))\nthis : 2 % (n + 3) = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 36
} | [
{
"pp": "case pos\nn : ℕ\nh1 : n ≠ 1\nh : IsCyclic (DihedralGroup n)\nh2 : n = 2\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 273,
"column": 8
} | {
"line": 274,
"column": 15
} | [
{
"pp": "n : ℕ\nhn : Odd n\nu : (ZMod n)ˣ := ZMod.unitOfCoprime 2 ⋯\nhu : ∀ (a : ZMod n), a + a = 0 ↔ a = 0\ni j : ZMod n\nh : Commute (r i, sr j).1 (r i, sr j).2\n⊢ (fun x ↦\n match x with\n | Sum.inl i => ⟨(sr i, r 0), ⋯⟩\n | Sum.inr (Sum.inl j) => ⟨(r 0, sr j), ⋯⟩\n | Sum.inr (Sum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 276,
"column": 8
} | {
"line": 277,
"column": 15
} | [
{
"pp": "n : ℕ\nhn : Odd n\nu : (ZMod n)ˣ := ZMod.unitOfCoprime 2 ⋯\nhu : ∀ (a : ZMod n), a + a = 0 ↔ a = 0\ni j : ZMod n\nh : Commute (sr i, r j).1 (sr i, r j).2\n⊢ (fun x ↦\n match x with\n | Sum.inl i => ⟨(sr i, r 0), ⋯⟩\n | Sum.inr (Sum.inl j) => ⟨(r 0, sr j), ⋯⟩\n | Sum.inr (Sum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 148,
"column": 8
} | {
"line": 149,
"column": 15
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\ninst✝³ : IsKleinFour G\nG₁ : Type u_2\nG₂ : Type u_3\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : IsKleinFour G₁\ne : G₁ ≃ G₂\nhe : e 1 = 1\nh : Monoid.exponent G₂ = 2\n_inst₁ : Fintype G₁ := Fintype.ofFinite G₁\n_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e\nx y : G₁\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 17
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝⁴ : Group G\ninst✝³ : IsKleinFour G\nG₁ : Type u_2\nG₂ : Type u_3\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : IsKleinFour G₁\ne : G₁ ≃ G₂\nhe : e 1 = 1\nh : Monoid.exponent G₂ = 2\n_inst₁ : Fintype G₁ := ⋯\n_inst₂ : Fintype G₂ := ⋯\nx y : G₁\nhx : e x ≠ 1\nhy : e y ≠ 1\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Congruence.BigOperators | {
"line": 33,
"column": 4
} | {
"line": 33,
"column": 50
} | [
{
"pp": "case nil\nι : Type u_1\nM : Type u_2\ninst✝ : MulOneClass M\nc : Con M\nf g : ι → M\nh : ∀ x ∈ [], c (f x) (g x)\n⊢ c (List.map f []).prod (List.map g []).prod",
"usedConstants": [
"MulOne.toOne",
"List.map",
"id",
"MulOne.toMul",
"MulOneClass.toMulOne",
"Con",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Congruence.BigOperators | {
"line": 43,
"column": 12
} | {
"line": 43,
"column": 23
} | [
{
"pp": "case mk\nι : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nc : Con M\ns : Multiset ι\nf g : ι → M\na✝ : List ι\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid ι)) a✝, c (f x) (g x)\n⊢ c (Multiset.map f (Quot.mk (⇑(List.isSetoid ι)) a✝)).prod (Multiset.map g (Quot.mk (⇑(List.isSetoid ι)) a✝)).prod",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 15
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Mul M\ninst✝ : Finite M\nh : Nonempty M\nthis : Fintype M\n⊢ commProb M = 1 ↔ IsMulCommutative M",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"instHDiv",
"congrArg",
"Rat",
"Commute",
"id",
"Subtype",
"HDiv.hDiv",
... | commProb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeGroup.IsFreeGroup | {
"line": 149,
"column": 38
} | {
"line": 149,
"column": 86
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nG✝ : Type u_3\nH : Type u_4\ninst✝² : Group G✝\ninst✝¹ : Group H\nG : Type u\ninst✝ : Group G\nX : Type u\nof : X → G\nlift : {H : Type u} → [inst : Group H] → (X → H) ≃ (G →* H)\nlift_of : ∀ {H : Type u} [inst : Group H] (f : X → H) (a : X), (lift f) (of a) = f a\n⊢ ∀ {H :... | by intro H _ f a; simp [← lift_of (lift.symm f)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.FreeGroup.IsFreeGroup | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : IsFreeGroup G\nH : Type u_2\ninst✝ : Group H\nf : Generators G → H\n⊢ ∃! F, ∀ (a : Generators G), F (of a) = f a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 15
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\n⊢ commProb G = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Rat",
"Commute",
"ConjClasses",
"id",
"MulOne.toMul",
"Subtype",
... | commProb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 93,
"column": 2
} | {
"line": 96,
"column": 33
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\n⊢ commProb G = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"mul_div_mul_right",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"HMul.hMu... | rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CommutingProbability | {
"line": 93,
"column": 2
} | {
"line": 96,
"column": 33
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\n⊢ commProb G = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"mul_div_mul_right",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"HMul.hMu... | rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CommutingProbability | {
"line": 107,
"column": 25
} | {
"line": 107,
"column": 73
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\np q : { p // Commute p.1 p.2 }\nh : (fun p ↦ ⟨(↑(↑p).1, ↑(↑p).2), ⋯⟩) p = (fun p ↦ ⟨(↑(↑p).1, ↑(↑p).2), ⋯⟩) q\n⊢ p = q",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subgroup.mul",
"Commute",
"Membership.me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 205,
"column": 15
} | {
"line": 205,
"column": 26
} | [
{
"pp": "case mul_of.inl\nM : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\none : C 1\ninl_mul : ∀ (m : M) (x : M ∗ N), C x → C (inl m * x)\ninr_mul : ∀ (n : N) (x : M ∗ N), C x → C (inr n * x)\nxs : FreeMonoid (M ⊕ N)\nih : C (mk xs)\nm : M\n⊢ C (mk (of (Sum.inl m) * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 206,
"column": 15
} | {
"line": 206,
"column": 26
} | [
{
"pp": "case mul_of.inr\nM : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\none : C 1\ninl_mul : ∀ (m : M) (x : M ∗ N), C x → C (inl m * x)\ninr_mul : ∀ (n : N) (x : M ∗ N), C x → C (inr n * x)\nxs : FreeMonoid (M ⊕ N)\nih : C (mk xs)\nn : N\n⊢ C (mk (of (Sum.inr n) * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 211,
"column": 22
} | {
"line": 211,
"column": 33
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\nm : M ∗ N\ninl : ∀ (m : M), C (Coprod.inl m)\ninr : ∀ (n : N), C (Coprod.inr n)\nmul : ∀ (x y : M ∗ N), C x → C y → C (x * y)\n⊢ C 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 137,
"column": 43
} | {
"line": 137,
"column": 54
} | [
{
"pp": "B : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B ≃ B'\n⊢ Surjective ⇑↑(FreeGroup.freeGroupCongr e)",
"usedConstants": [
"MulEquiv.instEquivLike",
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMulOneClass",
"MulEquiv.instMulEquivClass",
"id",
"MulOn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 383,
"column": 20
} | {
"line": 383,
"column": 64
} | [
{
"pp": "case cons\nB : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nx : B\nω' : List B\nih : cs.wordProd ω'.reverse = (cs.wordProd ω')⁻¹\n⊢ cs.wordProd (x :: ω').reverse = (cs.wordProd (x :: ω'))⁻¹",
"usedConstants": [
"Eq.mpr",
"CancelMonoid.toRightCanc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 421,
"column": 17
} | {
"line": 421,
"column": 46
} | [
{
"pp": "case succ\nB : Type u_1\nm : ℕ\nih : ∀ (i i' : B), (alternatingWord i i' m).length = m\ni i' : B\n⊢ (alternatingWord i i' (m + 1)).length = m + 1",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid",
"List.length_append",
"id... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 207,
"column": 61
} | {
"line": 207,
"column": 72
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nmotive : CoprodI M → Prop\none : motive 1\nmul : ∀ {i : ι} (m : M i) (x : CoprodI M), motive x → motive (of m * x)\nx : CoprodI M\nhx : x ∈ ⋃ i, range ⇑of\ny : CoprodI M\nihy : motive y\n⊢ ∃ i m, of m = x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 244,
"column": 55
} | {
"line": 253,
"column": 25
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_4\ninst✝¹ : (i : ι) → Group (G i)\nN : Type u_5\ninst✝ : Group N\nf : (i : ι) → G i →* N\ns : Subgroup N\nh : ∀ (i : ι), (f i).range ≤ s\n⊢ (lift f).range ≤ s",
"usedConstants": [
"Set.mem_range_self",
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"M... | by
rintro _ ⟨x, rfl⟩
induction x using CoprodI.induction_on with
| one => exact s.one_mem
| of i x =>
simp only [lift_of]
exact h i (Set.mem_range_self x)
| mul x y hx hy =>
simp only [map_mul]
exact s.mul_mem hx hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 459,
"column": 35
} | {
"line": 459,
"column": 46
} | [
{
"pp": "B : Type u_1\ni j : B\np k : ℕ\nh' : k < 2 * p → take k (alternatingWord i j (2 * p)) = if Even k then alternatingWord i j k else alternatingWord j i k\nh : k + 1 < 2 * p\nh_even : ¬Even k\nhk : take k (alternatingWord i j (2 * p)) = alternatingWord j i k\n⊢ Odd ?m.111",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 361,
"column": 6
} | {
"line": 362,
"column": 44
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\ninst✝ : (i : ι) → DecidableEq (M i)\ni : ι\nm : M i\nw : Word M\nh : w.fstIdx ≠ some i\nm' : M i\nw' : Word M\nh' : w'.fstIdx ≠ some i\nhe : rcons { head := m, tail := w, fstIdx_ne := h } = rcons { head := m', tail := w', fstIdx_ne := h' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 420,
"column": 32
} | {
"line": 420,
"column": 66
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\ni : ι\nw✝ : Word M\nj : ι\nm : M j\nw : Word M\nh1 : w.fstIdx ≠ some j\nh2 : m ≠ 1\nx✝ : { p // rcons p = w }\nij : ¬i = j\n⊢ (cons m w h1 h2).f... | by simp [cons, fstIdx, Ne.symm ij] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 47
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω₁ : List B\nhω₁ : cs.IsReduced ω₁\nω₂ : List B\nhω₂ : cs.IsReduced ω₂\nthis : cs.length (cs.wordProd (ω₁ ++ ω₂)) ≤ (ω₁ ++ ω₂).length\n⊢ cs.length (cs.wordProd ω₁ * cs.wordProd ω₂) ≤ cs.length (cs.wordProd ω₁) + c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 24
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw₁ w₂ : W\n⊢ cs.length w₂ ≤ cs.length (w₁ * w₂) + cs.length w₁",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"id",
"MulOne.toMul",
"D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw₁ w₂ : W\n⊢ cs.length w₁ ≤ cs.length (w₁ * w₂) + cs.length w₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 60
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw₁ w₂ : W\n⊢ ↑(cs.length (w₁ * w₂)) = ↑(cs.length w₁) + ↑(cs.length w₂)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 15
} | [
{
"pp": "case h₁\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\n⊢ cs.length (cs.simple i) ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.length (cs.simple i * w)⁻¹ ≠ cs.length w",
"usedConstants": [
"Eq.mpr",
"CoxeterSystem.inv_simple",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.toInvOne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 15
} | [
{
"pp": "case inl\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nh : cs.length (w * cs.simple i) + 1 ≤ cs.length w\n⊢ cs.length w ≤ cs.length (w * cs.simple i) + 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 15
} | [
{
"pp": "case inr\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nh : cs.length w + 1 ≤ cs.length (w * cs.simple i)\n⊢ cs.length (w * cs.simple i) ≤ cs.length w + 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 206,
"column": 60
} | {
"line": 206,
"column": 70
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nthis : cs.length (cs.simple i * w) = cs.length w⁻¹ + 1 ∨ cs.length (cs.simple i * w) + 1 = cs.length w⁻¹\n⊢ cs.length (cs.simple i * w) = cs.length w + 1 ∨ cs.length (cs.simple i * w) + 1 = cs.length... | length_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 15
} | [
{
"pp": "case step\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm✝¹ : ℕ\nhM : M.M i i' ≠ 0\nm✝ : ℕ\nm : (M.M i i').succ.le m✝\nih : cs.IsReduced (drop 1 ((if Even m✝ then i' else i) :: alternatingWord i i' m✝))\n⊢ cs.IsReduced (alternatingWord i i' m✝)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 13
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.IsRightDescent w⁻¹ i ↔ cs.IsLeftDescent w i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 319,
"column": 2
} | {
"line": 323,
"column": 7
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.IsRightDescent w i ↔ cs.length (w * cs.simple i) + 1 = cs.length w",
"usedConstants": [
"_private.Mathlib.GroupTheory.Coxeter.Length.0.CoxeterSystem.isRightDescent_iff._proof_1_2",
... | unfold IsRightDescent
constructor
· intro _
exact (cs.length_mul_simple w i).resolve_left (by lia)
· lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 319,
"column": 2
} | {
"line": 323,
"column": 7
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.IsRightDescent w i ↔ cs.length (w * cs.simple i) + 1 = cs.length w",
"usedConstants": [
"_private.Mathlib.GroupTheory.Coxeter.Length.0.CoxeterSystem.isRightDescent_iff._proof_1_2",
... | unfold IsRightDescent
constructor
· intro _
exact (cs.length_mul_simple w i).resolve_left (by lia)
· lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 29
} | [
{
"pp": "case mp\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw t : W\nh : cs.IsReflection (w * t * w⁻¹)\n⊢ cs.IsReflection t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 697,
"column": 42
} | {
"line": 697,
"column": 59
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nx y : (i : ι) × M i\nl : List ((i : ι) × M i)\nhnot1✝ : ∀ l_1 ∈ x :: y :: l, l_1.snd ≠ 1\nhnot1 : x.snd ≠ 1 ∧ ∀ x ∈ y :: l, x.snd ≠ 1\nhchain✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (x :: y :: l)\nhchain : x.fst ≠ y.fst ∧ List.IsChain ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 699,
"column": 6
} | {
"line": 699,
"column": 26
} | [
{
"pp": "case cons.cons\nι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nx : (i : ι) × M i\nl : List ((i : ι) × M i)\ni j : ι\nw' : NeWord M i j\nhnot1✝ : ∀ l_1 ∈ x :: ⟨i, w'.head⟩ :: l, l_1.snd ≠ 1\nhnot1 : x.snd ≠ 1 ∧ ∀ x ∈ ⟨i, w'.head⟩ :: l, x.snd ≠ 1\nhchain✝ : List.IsChain (fun l l' ↦ l.fst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 231,
"column": 8
} | {
"line": 231,
"column": 19
} | [
{
"pp": "case cons\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\nω : List B\nih : cs.leftInvSeq ω = (cs.rightInvSeq ω.reverse).reverse\n⊢ cs.leftInvSeq (i :: ω) = (cs.rightInvSeq (i :: ω).reverse).reverse",
"usedConstants": [
"Eq.mpr",
"MulEqui... | leftInvSeq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CoprodI | {
"line": 816,
"column": 27
} | {
"line": 816,
"column": 38
} | [
{
"pp": "case singleton\nι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i\ni j i✝ : ι\nx : H i✝\nhne_one : x ≠ 1\nk : ι\nhk... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 829,
"column": 25
} | {
"line": 829,
"column": 43
} | [
{
"pp": "ι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 869,
"column": 8
} | {
"line": 869,
"column": 94
} | [
{
"pp": "case neg\nι : Type u_1\nG : Type u_4\ninst✝³ : Group G\nH : ι → Type u_5\ninst✝² : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝¹ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 →... | exact lift_word_prod_nontrivial_of_head_card f X hXnonempty hXdisj hpp w hcard hl.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.CoprodI | {
"line": 876,
"column": 8
} | {
"line": 876,
"column": 19
} | [
{
"pp": "case pos\nι : Type u_1\nG : Type u_4\ninst✝³ : Group G\nH : ι → Type u_5\ninst✝² : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝¹ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 1032,
"column": 8
} | {
"line": 1032,
"column": 19
} | [
{
"pp": "case H\nι : Type u_1\ninst✝² : Nontrivial ι\nG : Type u_1\ninst✝¹ : Group G\na : ι → G\nα : Type u_4\ninst✝ : MulAction G α\nX Y : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhYdisj : Pairwise (Disjoint on Y)\nhXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)\nhX : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 420,
"column": 50
} | {
"line": 420,
"column": 61
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < (cs.rightInvSeq ω).length\ndup : (cs.rightInvSeq ω)[j]? = (cs.rightInvSeq ω)[j']?\n⊢ j' < ω.length",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 452,
"column": 51
} | {
"line": 452,
"column": 62
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni j : B\np k : ℕ\nh : k + 1 < 2 * p\n⊢ k + 1 < (cs.leftInvSeq (alternatingWord i j (2 * p))).length",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"instMulNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 17
} | [
{
"pp": "case a\nB✝ : Type u_1\ninst✝⁷ : TopologicalSpace B✝\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B✝ → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B✝) → NormedAddCommGroup (E x)\ninst✝² : (x : B✝) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 23
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na b : Quotient ↑H ↑K\nh : ¬Disjoint (doubleCoset (Quotient.out a) ↑H ↑K) (doubleCoset (Quotient.out b) ↑H ↑K)\n⊢ a = b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 45
} | [
{
"pp": "case h.mp\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na x : G\ny : ↥K\nh_h : x * ((↑y)⁻¹ * a⁻¹) ∈ H\n⊢ ∃ x_1 ∈ H, ∃ y ∈ K, x = x_1 * a * y",
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"Group.toDivisionMonoid",
"Member... | refine ⟨x * (y⁻¹ * a⁻¹), h_h, y, y.2, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.DoubleCoset | {
"line": 214,
"column": 27
} | {
"line": 214,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na✝ : Finite (Quotient ↑H ↑K)\nval✝ : Fintype (Quotient ↑H ↑K)\n⊢ ⋃ i ∈ Finset.univ, quotToDoubleCoset H K i = Set.univ",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"Iff.of_eq",
"congrArg",
"Finset",
"Set.univ",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 38,
"column": 4
} | {
"line": 39,
"column": 11
} | [
{
"pp": "ι : Type u_1\nG : Type u_2\ninst✝¹ : Finite ι\ninst✝ : Monoid G\nn : ι → ℕ\ne : G ≃* ((i : ι) → Multiplicative (ZMod (n i)))\ni : ι\n⊢ n i = orderOf (e.symm (Pi.mulSingle i (Multiplicative.ofAdd 1)))",
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
"Multiplicative.monoid",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 57
} | [
{
"pp": "case mpr.mk\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nI : Finset (Quotient ↑H ↑K)\nhI : ∀ (x : G), x ∈ ⋃ i ∈ I, quotToDoubleCoset H K i\nx✝ : Quotient ↑H ↑K\ng : G\ni : Quotient ↑H ↑K\nhi : i ∈ I\nhT : g ∈ quotToDoubleCoset H K i\n⊢ Quot.mk (⇑(setoid ↑H ↑K)) g ∈ ↑I",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 231,
"column": 55
} | {
"line": 231,
"column": 66
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nx : Quot ⇑(leftRel K)\ny : G\nhy : ⟦y⟧ = x\ncover : ¬∃ i, y ∈ doubleCoset (out i) ↑H ↑K\n⊢ y ∉ ⋃ q, quotToDoubleCoset H K q",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"congrArg",
"Set.mem_iUnion._simp_1",
"Memb... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 90
} | [
{
"pp": "case h\nG : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nH : ∀ (n : ℕ), n ∣ Monoid.exponent G → ∀ (a : ZMod n), a ≠ 0 → ∃ φ, φ (Multiplicative.ofAdd a) ≠ 1\na : G\nha : a ≠ 1\nι : Type\nw✝ : Fintype ι\nn : ι → ℕ\nleft✝ : ∀ (i : ι), 1 < n i\nh : Nonempty (G ≃* (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 60
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\ng g' : G\nh : ∀ (φ : G →* Mˣ), φ g = φ g'\n⊢ g = g'",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 90,
"column": 6
} | {
"line": 91,
"column": 13
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nι : Type\nw✝ : Fintype ι\nn : ι → ℕ\nh₁ : ∀ (i : ι), 1 < n i\nh₂ : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))\ne : G ≃* ((i : ι) → Multiplicative (ZM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 242,
"column": 55
} | {
"line": 242,
"column": 66
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nx : Quot ⇑(rightRel H)\ny : G\nhy : ⟦y⟧ = x\ncover : ¬∃ i, y ∈ doubleCoset (out i) ↑H ↑K\n⊢ y ∉ ⋃ q, quotToDoubleCoset H K q",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"congrArg",
"Set.mem_iUnion._simp_1",
"Mem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 278,
"column": 46
} | {
"line": 278,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nt : Finset (Quotient ↑H ↑K)\nht : ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K ≠ Set.univ\nx : G\ny : Quotient ↑H ↑K\nhy : y ∈ t\nq : G\nhq : q ∈ doubleCoset (out y) ↑H ↑K\nhx : Quot.mk (⇑(rightRel H)) q = Quot.mk (⇑(rightRel H)) x\na : ↥H\nha : x = ↑a * q\n⊢ x = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteIndexNormalSubgroup | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 17
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\n⊢ Subgroup.comap f K.toSubgroup = g.ker",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteIndexNormalSubgroup | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 22
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\nhker : Subgroup.comap f K.toSubgroup = g.ker\n⊢ (Subgroup.comap f K.toSubgroup).FiniteIndex",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FixedPointFree | {
"line": 59,
"column": 31
} | {
"line": 59,
"column": 60
} | [
{
"pp": "F : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : FunLike F G G\ninst✝¹ : MonoidHomClass F G G\nφ : F\ninst✝ : Finite G\nhφ : FixedPointFree ⇑φ\nh2 : (⇑φ)^[2] = _root_.id\ng : G\n⊢ g * φ g = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 15
} | [
{
"pp": "case inl.e_m.e_a\nM : Type u_2\ninst✝ : AddCommGroup M\na : ℚ\nm : M\ns : ℕ+\nh : 0 ≤ a\n⊢ ↑(have this := ⟨a, h⟩;\n this).num =\n a.num",
"usedConstants": [
"Int.instAddCommGroup",
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Int.instIsStrictOrderedRing",
"I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 17
} | [
{
"pp": "case e_a\nM : Type u_2\ninst✝ : AddCommGroup M\na : ℚ\nm : M\ns : ℕ+\nh : a ≤ 0\n⊢ ↑a.num.natAbs = -a.num",
"usedConstants": [
"Int.instAddCommGroup",
"abs_eq_neg_self._simp_1",
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Int.instIsStrictOrderedRing",
"Int.cast... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 322,
"column": 38
} | {
"line": 322,
"column": 49
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\na : ℚ≥0\nha : 0 < a\nmb : M\nsb : ℕ+\nmc : M\nsc : ℕ+\nh : ↑sc • mb < ↑sb • mc\n⊢ ↑⟨a.den, ⋯⟩ * a.num ≠ 0",
"usedConstants": [
"PNat.val",
"Eq.mp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 329,
"column": 6
} | {
"line": 329,
"column": 17
} | [
{
"pp": "case mk.refine_2\nM✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nb c : ℚ≥0\nh : b < c\nm : M\ns : ℕ+\nha : ↑s • 0 < ↑1 • m\n⊢ 0 < m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 339,
"column": 41
} | {
"line": 339,
"column": 52
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\na : ℚ\nha : 0 < a\nb c : DivisibleHull M\nh : b < c\n⊢ 0 < ⟨a, ⋯⟩",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 344,
"column": 23
} | {
"line": 344,
"column": 52
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\na : DivisibleHull M\nha : 0 < a\nb c : ℚ\nh : b < c\n⊢ 0 < ⟨c - b, ⋯⟩",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 351,
"column": 24
} | {
"line": 351,
"column": 35
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\na b : M\nh : a ≤ b\n⊢ (↑(coeAddMonoidHom M)).toFun a ≤ (↑(coeAddMonoidHom M)).toFun b",
"usedConstants": [
"PNat.val",
"Eq.mpr",
"instHSMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 391,
"column": 6
} | {
"line": 391,
"column": 45
} | [
{
"pp": "case a\nM✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\nx : DivisibleHull M\nx✝³ x✝² : M\nx✝¹ x✝ : ℕ+\nh : ArchimedeanClass.mk (mk x✝³ x✝¹) = ArchimedeanClass.mk (mk x✝² x✝)\n⊢ (archimedeanClassOrderHom M) ((fun m s ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 396,
"column": 6
} | {
"line": 396,
"column": 17
} | [
{
"pp": "case mk.mk\nM✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\nnum✝¹ : M\nden✝¹ : ℕ+\nnum✝ : M\nden✝ : ℕ+\nh : (archimedeanClassOrderHom M) (ArchimedeanClass.mk num✝¹) ≤ (archimedeanClassOrderHom M) (ArchimedeanClass.mk... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 100,
"column": 27
} | {
"line": 100,
"column": 38
} | [
{
"pp": "case mul\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ y✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nhy✝ : y✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIHa : g * x✝ * g⁻¹ ∈ H.focalSubgroup\nIHb : g * y✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * (x✝ * y✝) * g⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 101,
"column": 18
} | {
"line": 101,
"column": 41
} | [
{
"pp": "case inv\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIH : g * x✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * x✝⁻¹ * g⁻¹ ∈ H.focalSubgroup",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Transfer | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 84
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nhH : ¬H.index = 0\nthis : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH\nf : Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=\n fun q ↦ ⟨g, ⋯⟩ ^ Function.minimalPeriod (fun x ↦ g • x) q.out\nhf : ∀ (q : Quotient (orbitRel (↥(zpow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 13
} | [
{
"pp": "case h\nG : Type u_1\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : (↑P).FiniteIndex\ng : ↥P\nhQ : IsPGroup p (↥↑P ⧸ (↑P).focalSubgroupOf)\n⊢ ↑g ^ (↑P).index = 1 ↔ g ∈ (↑P).focalSubgroupOf",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Radical | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 15
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : IsCoatomic α\na : α\nh : a ⊔ radical α = ⊤\nm : α\nc : IsCoatom m\nle : a ≤ m\nq : m = ⊤\n⊢ a = ⊤",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 76,
"column": 16
} | {
"line": 76,
"column": 39
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\nn✝ n : ℕ\nhead : α × Bool\ntail : List (α × Bool)\nh : IsCyclicallyReduced (head :: tail)\n⊢ ∀ l ∈ replicate (n + 1) (head :: tail), IsChain (fun a b ↦ a.1 = b.1 → a.2 = b.2) l",
"usedConstants": [
"Eq.mpr",
"List.replicate",
"False",
"Nat.in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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