module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 75
} | {
"line": 401,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | [] | simp [isBoundaryPoint_iff_not_isInteriorPoint, hf.isInteriorPoint_iff hn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.Hom | {
"line": 359,
"column": 2
} | {
"line": 360,
"column": 47
} | {
"line": 362,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF₁ : Type u_3\nF₂ : Type u_4\nM : Type u_6\ninst✝²⁸ : NontriviallyNormedField 𝕜\nE₁ : B → Type u_7\ninst✝²⁷ : (x : B) → AddCommGroup (E₁ x)\ninst✝²⁶ : (x : B) → Module 𝕜 (E₁ x)\ninst✝²⁵ : NormedAddCommGroup F₁\ninst✝²⁴ : NormedSpace 𝕜 F₁\ninst✝²³ : TopologicalSpace (Tota... | [] | simp only [mdifferentiableWithinAt_hom_bundle] at hϕ
exact hϕ.2.clm_apply_of_inCoordinates hv hϕ.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.Hom | {
"line": 359,
"column": 2
} | {
"line": 360,
"column": 47
} | {
"line": 362,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF₁ : Type u_3\nF₂ : Type u_4\nM : Type u_6\ninst✝²⁸ : NontriviallyNormedField 𝕜\nE₁ : B → Type u_7\ninst✝²⁷ : (x : B) → AddCommGroup (E₁ x)\ninst✝²⁶ : (x : B) → Module 𝕜 (E₁ x)\ninst✝²⁵ : NormedAddCommGroup F₁\ninst✝²⁴ : NormedSpace 𝕜 F₁\ninst✝²³ : TopologicalSpace (Tota... | [] | simp only [mdifferentiableWithinAt_hom_bundle] at hϕ
exact hϕ.2.clm_apply_of_inCoordinates hv hϕ.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 81
} | {
"line": 96,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nm n : WithTop ℕ∞\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\nE' : Type u_5... | [
"𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nm n : WithTop ℕ∞\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\nE' : Type u_5\ninst✝⁹ : N... | have : MapsTo (fun x ↦ (x, g x)) t (t ×ˢ u) := fun y hy ↦ by simp [hy, hu hy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 35
} | {
"line": 188,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns t : Set M\nx : M\nV W : (x : M) → Ta... | [
"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns t : Set M\nx : M\nV W : (x : M) → TangentSpace I... | apply mlieBracketWithin_congr_set | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 119,
"column": 14
} | {
"line": 119,
"column": 16
} | {
"line": 120,
"column": 6
} | [
{
"pp": "case hst\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nm n : WithTop ℕ∞\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\nE' ... | [
"case hst\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nm n : WithTop ℕ∞\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\nE' : Type u_5\n... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 122,
"column": 8
} | {
"line": 122,
"column": 39
} | {
"line": 123,
"column": 6
} | [
{
"pp": "case hst\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nm n : WithTop ℕ∞\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\nE' ... | [] | exact hu (inter_subset_left ha) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 122,
"column": 8
} | {
"line": 122,
"column": 39
} | {
"line": 123,
"column": 6
} | [
{
"pp": "case hst\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nm n : WithTop ℕ∞\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\nE' ... | [] | exact hu (inter_subset_left ha) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 122,
"column": 8
} | {
"line": 122,
"column": 39
} | {
"line": 123,
"column": 6
} | [
{
"pp": "case hst\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nm n : WithTop ℕ∞\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\nE' ... | [] | exact hu (inter_subset_left ha) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 55
} | {
"line": 526,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nH' : Type u_5\ninst✝⁷ : Topologic... | [
"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nH' : Type u_5\ninst✝⁷ : TopologicalSpace H'\n... | simp only [mpullbackWithin, Bundle.TotalSpace.mk_inj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 35
} | {
"line": 438,
"column": 0
} | [
{
"pp": "x y : ℝ\nhxy : Fact (x < y)\nz : ↑(Icc x y)\nh : ↑z < y\n⊢ chartAt (EuclideanHalfSpace 1) z = IccLeftChart x y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"IccRightChart",
"chartAt",
"congrArg",
"EuclideanHalfSpace",
"Real.decidableL... | [] | simp [Icc_chartedSpaceChartAt, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 35
} | {
"line": 438,
"column": 0
} | [
{
"pp": "x y : ℝ\nhxy : Fact (x < y)\nz : ↑(Icc x y)\nh : ↑z < y\n⊢ chartAt (EuclideanHalfSpace 1) z = IccLeftChart x y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"IccRightChart",
"chartAt",
"congrArg",
"EuclideanHalfSpace",
"Real.decidableL... | [] | simp [Icc_chartedSpaceChartAt, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 35
} | {
"line": 438,
"column": 0
} | [
{
"pp": "x y : ℝ\nhxy : Fact (x < y)\nz : ↑(Icc x y)\nh : ↑z < y\n⊢ chartAt (EuclideanHalfSpace 1) z = IccLeftChart x y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"IccRightChart",
"chartAt",
"congrArg",
"EuclideanHalfSpace",
"Real.decidableL... | [] | simp [Icc_chartedSpaceChartAt, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Instances.Icc | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 12
} | {
"line": 89,
"column": 13
} | [
{
"pp": "case pos\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ↑(Icc x y)\nφ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) := ⋯\nφ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := ⋯\nhz : ↑z < y\n⊢ EqOn\n (↑((Homeomorph.addLeft (-x)).toOpenPartialHomeomorph.extend 𝓘(ℝ, ℝ)) ∘\n (f... | [
"case pos\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ↑(Icc x y)\nφ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) :=\n ContinuousLinearEquiv.prodUnique ℝ (EuclideanSpace ℝ (Fin 1)) Unit\nφ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := φ₀.trans (PiLp.equivOfUnique 2 ℝ fun x ↦ ℝ)\nhz : ↑z ... | intro z' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Geometry.Manifold.Instances.Icc | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 12
} | {
"line": 108,
"column": 13
} | [
{
"pp": "case neg\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ↑(Icc x y)\nφ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) := ⋯\nφ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := ⋯\nhz : ¬↑z < y\n⊢ EqOn\n (↑((Homeomorph.pointReflection (y / 2)).toOpenPartialHomeomorph.extend 𝓘(ℝ, ℝ))... | [
"case neg\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ↑(Icc x y)\nφ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) :=\n ContinuousLinearEquiv.prodUnique ℝ (EuclideanSpace ℝ (Fin 1)) Unit\nφ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := φ₀.trans (PiLp.equivOfUnique 2 ℝ fun x ↦ ℝ)\nhz : ¬↑z... | intro z' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Geometry.Manifold.IntegralCurve.Basic | {
"line": 175,
"column": 2
} | {
"line": 185,
"column": 5
} | {
"line": 187,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\ns : Set ℝ\nt₀ : ℝ\ninst✝ : IsManifold I 1 M\nhγ ... | [] | replace hsrc := extChartAt_source I (γ t₀) ▸ hsrc
rw [hasDerivWithinAt_iff_hasFDerivWithinAt, ← hasMFDerivWithinAt_iff_hasFDerivWithinAt]
apply (HasMFDerivWithinAt.comp t (hasMFDerivWithinAt_extChartAt (I := I) hsrc) (hγ _ ht)
(Set.subset_preimage_image _ _)).congr_mfderiv
rw [ContinuousLinearMap.ext_iff]
i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IntegralCurve.Basic | {
"line": 175,
"column": 2
} | {
"line": 185,
"column": 5
} | {
"line": 187,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\ns : Set ℝ\nt₀ : ℝ\ninst✝ : IsManifold I 1 M\nhγ ... | [] | replace hsrc := extChartAt_source I (γ t₀) ▸ hsrc
rw [hasDerivWithinAt_iff_hasFDerivWithinAt, ← hasMFDerivWithinAt_iff_hasFDerivWithinAt]
apply (HasMFDerivWithinAt.comp t (hasMFDerivWithinAt_extChartAt (I := I) hsrc) (hγ _ ht)
(Set.subset_preimage_image _ _)).congr_mfderiv
rw [ContinuousLinearMap.ext_iff]
i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Instances.Sphere | {
"line": 235,
"column": 98
} | {
"line": 245,
"column": 6
} | {
"line": 247,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv : E\nhv : ‖v‖ = 1\nw : ↥(ℝ ∙ v)ᗮ\n⊢ stereoToFun v ↑(stereoInvFun hv w) = w",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"one_pow",
"Real.instIsOrderedR... | [] | by
simp only [stereoToFun, stereoInvFun, stereoInvFunAux, smul_add, map_add, map_smul,
innerSL_apply_apply, Submodule.orthogonalProjectionOnto_mem_subspace_eq_self]
have h₁ : (ℝ ∙ v)ᗮ.orthogonalProjectionOnto v = 0 :=
Submodule.orthogonalProjectionOnto_orthogonalComplement_singleton_eq_zero v
have h₂ : ⟪v... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 95,
"column": 49
} | {
"line": 95,
"column": 51
} | {
"line": 95,
"column": 52
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : (x : M) → ENorm (TangentSpace I x)\na✝ b : ℝ\nγ γ' : ℝ → M\nh : EqOn γ γ' (Ioo a✝ b... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : (x : M) → ENorm (TangentSpace I x)\na✝ b : ℝ\nγ γ' : ℝ → M\nh : EqOn γ γ' (Ioo a✝ b)\nt : ℝ\nht... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Manifold.Riemannian.Basic | {
"line": 163,
"column": 4
} | {
"line": 164,
"column": 77
} | {
"line": 165,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nn : ℕ∞ω\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSp... | [
"case refine_1\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nn : ℕ∞ω\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y... | have D : ContDiffOn ℝ 1 e (Icc 0 1) :=
contMDiffOn_iff_contDiffOn.mp (hγ.comp_contMDiffOn contMDiffOn_projIcc) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.Riemannian.Basic | {
"line": 169,
"column": 42
} | {
"line": 169,
"column": 67
} | {
"line": 169,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nn : ℕ∞ω\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : ... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nn : ℕ∞ω\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nγ : Path ... | show x = e 0 by simp [e], | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Geometry.Manifold.Sheaf.Smooth | {
"line": 233,
"column": 16
} | {
"line": 236,
"column": 43
} | {
"line": 236,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\ninst✝²⁸ : NontriviallyNormedField 𝕜\nEM : Type u_2\ninst✝²⁷ : NormedAddCommGroup EM\ninst✝²⁶ : NormedSpace 𝕜 EM\nHM : Type u_3\ninst✝²⁵ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\nE : Type u_4\ninst✝²⁴ : NormedAddCommGroup E\ninst✝²³ : NormedSpace 𝕜 E\nH : Type u_5\ninst✝²²... | [] | by
rw [CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget _ _
(CategoryTheory.forget CommGrpCat)]
exact (smoothSheaf IM I M A).property | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 916,
"column": 2
} | {
"line": 917,
"column": 66
} | {
"line": 919,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁷ : TopologicalSpace H\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : IsManifold I (minSmoothness ... | [] | simp only [← contMDiffOn_univ] at hU hV ⊢
exact hU.mlieBracketWithin_vectorField hV uniqueMDiffOn_univ hmn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 916,
"column": 2
} | {
"line": 917,
"column": 66
} | {
"line": 919,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁷ : TopologicalSpace H\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : IsManifold I (minSmoothness ... | [] | simp only [← contMDiffOn_univ] at hU hV ⊢
exact hU.mlieBracketWithin_vectorField hV uniqueMDiffOn_univ hmn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Metric | {
"line": 139,
"column": 2
} | {
"line": 140,
"column": 69
} | {
"line": 142,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace ℝ E\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nF : Type u_4\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\nV : M → Type u_... | [] | unfold derivMetricTensor
rw [TensorialAt.mkHom₂_apply _ _ hσ hτ, derivMetricTensorAux_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Metric | {
"line": 139,
"column": 2
} | {
"line": 140,
"column": 69
} | {
"line": 142,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace ℝ E\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nF : Type u_4\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\nV : M → Type u_... | [] | unfold derivMetricTensor
rw [TensorialAt.mkHom₂_apply _ _ hσ hτ, derivMetricTensorAux_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Commensurable | {
"line": 90,
"column": 66
} | {
"line": 90,
"column": 85
} | {
"line": 90,
"column": 86
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : ConjAct G\n⊢ (g • H).Commensurable H ↔ H.Commensurable (g⁻¹ • H)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Subgroup.Commensur... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : ConjAct G\n⊢ (?m.21 • g • H).Commensurable (?m.21 • H) ↔ H.Commensurable (g⁻¹ • H)",
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : ConjAct G\n⊢ ConjAct G"
] | commensurable_conj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 217,
"column": 40
} | {
"line": 217,
"column": 49
} | {
"line": 217,
"column": 49
} | [
{
"pp": "case inr.a.r\nn : ℕ\nhn : NeZero n\nm : ZMod n\n⊢ orderOf (r m) ∣ lcm n 2",
"ppTerm": "?inr.a.r",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"DihedralGroup.orderOf_r",
"Eq.mpr",
"Dvd.dvd",
"instHDiv",
"congrArg",
"DihedralGroup.instGroup",
... | [
"case inr.a.r\nn : ℕ\nhn : NeZero n\nm : ZMod n\n⊢ n / n.gcd m.val ∣ lcm n 2"
] | orderOf_r | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 21
} | {
"line": 106,
"column": 22
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\n⊢ ↑(Nat.card { p // Commute p.1 p.2 }) ≤\n ↑(Nat.card { p // Commute p.1 p.2 }) / ↑(Nat.card G) ^ 2 * ↑(H.index * Nat.card ↥H) ^ 2",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSe... | [
"G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\n⊢ ↑(Nat.card { p // Commute p.1 p.2 }) ≤\n ↑(Nat.card { p // Commute p.1 p.2 }) / ↑(Nat.card G) ^ 2 * ↑(Nat.card ↥H * H.index) ^ 2",
"G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\n⊢ 0 < ↑(Nat.card ↥H) ^ 2"
] | mul_comm H.index, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 108,
"column": 25
} | {
"line": 108,
"column": 75
} | {
"line": 109,
"column": 2
} | [
{
"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",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoi... | [] | simpa only [Subtype.ext_iff, Prod.ext_iff] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.CommutingProbability | {
"line": 108,
"column": 25
} | {
"line": 108,
"column": 75
} | {
"line": 109,
"column": 2
} | [
{
"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",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoi... | [] | simpa only [Subtype.ext_iff, Prod.ext_iff] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CommutingProbability | {
"line": 108,
"column": 25
} | {
"line": 108,
"column": 75
} | {
"line": 109,
"column": 2
} | [
{
"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",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoi... | [] | simpa only [Subtype.ext_iff, Prod.ext_iff] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CommutingProbability | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 75
} | {
"line": 109,
"column": 2
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\n⊢ Nat.card { p // Commute p.1 p.2 } ≤ Nat.card { p // Commute p.1 p.2 }",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.card_le_card_of_injective",
"HMul.hMul",
"Monoid.toMulOn... | [
"case h\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\n⊢ ↑(Nat.card G) ^ 2 ≠ 0",
"G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\n⊢ 0 < ↑(Nat.card ↥H) ^ 2"
] | · refine Nat.card_le_card_of_injective (fun p ↦ ⟨⟨p.1.1, p.1.2⟩, Subtype.ext_iff.mp p.2⟩) ?_
exact fun p q h ↦ by simpa only [Subtype.ext_iff, Prod.ext_iff] using h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 299,
"column": 8
} | {
"line": 299,
"column": 81
} | {
"line": 299,
"column": 81
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nM' : Type u_3\nN' : Type u_4\nP : Type u_5\ninst✝⁴ : MulOneClass M\ninst✝³ : MulOneClass N\ninst✝² : MulOneClass M'\ninst✝¹ : MulOneClass N'\ninst✝ : MulOneClass P\nf : M →* M'\ng : N →* N'\n⊢ (mk.comp (FreeMonoid.map (Sum.map ⇑f ⇑g))) (of (Sum.inl 1)) = 1",
"ppTerm": "?... | [] | simp only [MonoidHom.comp_apply, map_of, Sum.map_inl, map_one, mk_of_inl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 299,
"column": 8
} | {
"line": 299,
"column": 81
} | {
"line": 299,
"column": 81
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nM' : Type u_3\nN' : Type u_4\nP : Type u_5\ninst✝⁴ : MulOneClass M\ninst✝³ : MulOneClass N\ninst✝² : MulOneClass M'\ninst✝¹ : MulOneClass N'\ninst✝ : MulOneClass P\nf : M →* M'\ng : N →* N'\n⊢ (mk.comp (FreeMonoid.map (Sum.map ⇑f ⇑g))) (of (Sum.inl 1)) = 1",
"ppTerm": "?... | [] | simp only [MonoidHom.comp_apply, map_of, Sum.map_inl, map_one, mk_of_inl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 299,
"column": 8
} | {
"line": 299,
"column": 81
} | {
"line": 299,
"column": 81
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nM' : Type u_3\nN' : Type u_4\nP : Type u_5\ninst✝⁴ : MulOneClass M\ninst✝³ : MulOneClass N\ninst✝² : MulOneClass M'\ninst✝¹ : MulOneClass N'\ninst✝ : MulOneClass P\nf : M →* M'\ng : N →* N'\n⊢ (mk.comp (FreeMonoid.map (Sum.map ⇑f ⇑g))) (of (Sum.inl 1)) = 1",
"ppTerm": "?... | [] | simp only [MonoidHom.comp_apply, map_of, Sum.map_inl, map_one, mk_of_inl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.PresentedGroup | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 65
} | {
"line": 97,
"column": 0
} | [
{
"pp": "α : Type u_1\nrels : Set (FreeGroup α)\nthis : of = ⇑(QuotientGroup.mk' (Subgroup.normalClosure rels)) ∘ FreeGroup.of\n⊢ (QuotientGroup.mk' (Subgroup.normalClosure rels)).range = ⊤",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MonoidHom.range",
"Mon... | [] | exact MonoidHom.range_eq_top.2 (QuotientGroup.mk'_surjective _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.CoprodI | {
"line": 326,
"column": 41
} | {
"line": 326,
"column": 63
} | {
"line": 328,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\ni : ι\nm : M i\nw : Word M\nhmw : w.fstIdx ≠ some i\nh1 : m ≠ 1\n⊢ (cons m w hmw h1).fstIdx = some i",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Option.some",
"eq_self",
"of_eq_true",
"Eq"... | [] | by simp [cons, fstIdx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 497,
"column": 11
} | {
"line": 497,
"column": 26
} | {
"line": 497,
"column": 27
} | [
{
"pp": "B : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm : ℕ\nhm : m ≤ M.M i i' * 2\n⊢ (if Even ↑m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2) =\n (if Even ↑(M.M i i' * 2 - m) then 1 else cs.simple i) * (cs.simple i' * cs.simple i) ... | [
"B : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm : ℕ\nhm : m ≤ M.M i i' * 2\n⊢ (if Even ↑m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ↑(m / 2) =\n (if Even ↑(M.M i i' * 2 - m) then 1 else cs.simple i) * (cs.simple i' * cs.simple i) ^ ↑((M.M i ... | ← zpow_natCast, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 334,
"column": 2
} | {
"line": 335,
"column": 7
} | {
"line": 337,
"column": 0
} | [
{
"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.IsLeftDescent w i ↔ ¬cs.IsLeftDescent (cs.simple i * w) i",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
... | [] | rw [isLeftDescent_iff, not_isLeftDescent_iff, simple_mul_simple_cancel_left]
tauto | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 334,
"column": 2
} | {
"line": 335,
"column": 7
} | {
"line": 337,
"column": 0
} | [
{
"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.IsLeftDescent w i ↔ ¬cs.IsLeftDescent (cs.simple i * w) i",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
... | [] | rw [isLeftDescent_iff, not_isLeftDescent_iff, simple_mul_simple_cancel_left]
tauto | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 338,
"column": 62
} | {
"line": 340,
"column": 7
} | {
"line": 342,
"column": 0
} | [
{
"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.IsRightDescent (w * cs.simple i) i",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",... | [] | by
rw [isRightDescent_iff, not_isRightDescent_iff, simple_mul_simple_cancel_right]
tauto | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Descent | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 89
} | {
"line": 94,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝¹ : Group G\nf : G →* G\nhf : ∀ (U : Subgroup G), map f U ≤ U\ns : Set G\nh : G → ℝ\na b c : ℝ\nha : 0 ≤ a\nH₀ : a < b\nhs : s.Finite\nH₁ : s * ↑f.range = Set.univ\nH₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c\nH₃ : ∀ (x : G), b * h x - c ≤ h (f x)\ninst✝ : Northcott h\... | [] | exact hx₁ <| U.mul_mem (mem_closure_of_mem <| .inl hg) <| hf U <| mem_map_of_mem f hy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Descent | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 89
} | {
"line": 94,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝¹ : Group G\nf : G →* G\nhf : ∀ (U : Subgroup G), map f U ≤ U\ns : Set G\nh : G → ℝ\na b c : ℝ\nha : 0 ≤ a\nH₀ : a < b\nhs : s.Finite\nH₁ : s * ↑f.range = Set.univ\nH₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c\nH₃ : ∀ (x : G), b * h x - c ≤ h (f x)\ninst✝ : Northcott h\... | [] | exact hx₁ <| U.mul_mem (mem_closure_of_mem <| .inl hg) <| hf U <| mem_map_of_mem f hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Descent | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 89
} | {
"line": 94,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝¹ : Group G\nf : G →* G\nhf : ∀ (U : Subgroup G), map f U ≤ U\ns : Set G\nh : G → ℝ\na b c : ℝ\nha : 0 ≤ a\nH₀ : a < b\nhs : s.Finite\nH₁ : s * ↑f.range = Set.univ\nH₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c\nH₃ : ∀ (x : G), b * h x - c ≤ h (f x)\ninst✝ : Northcott h\... | [] | exact hx₁ <| U.mul_mem (mem_closure_of_mem <| .inl hg) <| hf U <| mem_map_of_mem f hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.DivisibleHull | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 69
} | {
"line": 260,
"column": 4
} | [
{
"pp": "case e'_1.e'_3\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nm n m' n' : M\ns t s' t' : ℕ+\nh : ↑s' • m = ↑s • m'\nh' : ↑t' • n = ↑t • n'\n⊢ (↑s' * ↑t') • ↑t • m = (↑s * ↑t) • ↑t' • m'",
"ppTerm": "?e'_1.e'_3",
"assigned": true,
"usedCo... | [
"case e'_1.e'_3\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nm n m' n' : M\ns t s' t' : ℕ+\nh : ↑s' • m = ↑s • m'\nh' : ↑t' • n = ↑t • n'\n⊢ (↑t' * (↑t * ↑s)) • m' = (↑s * (↑t * ↑t')) • m'"
] | simp_rw [smul_smul, mul_rotate s'.val, ← smul_smul, h, smul_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.DoubleCoset | {
"line": 198,
"column": 55
} | {
"line": 202,
"column": 34
} | {
"line": 204,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ Quotient ↑⊥ ↑H = (G ⧸ H)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"congrArg",
"QuotientGroup.leftRel",
"Eq.rec",
"id",
"QuotientGroup.instHasQuotientSubgroup",
"Subgroup",
"Setoid",
... | [] | by
unfold Quotient
congr
ext
simp_rw [← bot_rel_eq_leftRel H] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.CoprodI | {
"line": 1035,
"column": 8
} | {
"line": 1035,
"column": 23
} | {
"line": 1036,
"column": 6
} | [
{
"pp": "ι : 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 : ∀ (i : ι)... | [] | simpa using hgt | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.Transfer | {
"line": 131,
"column": 42
} | {
"line": 131,
"column": 71
} | {
"line": 131,
"column": 72
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)\nk : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))\n⊢ g •\n (g ^ ((H.quotientEquivSigmaZMod g) (g ^ (-1 + k.cast) • Quotient.out q)).snd.cast *\n Quotient.out (Quotient.out ((H.quotien... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)\nk : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))\n⊢ g • (g ^ ⟨q, ↑(-1 + k.cast)⟩.snd.cast * Quotient.out (Quotient.out ⟨q, ↑(-1 + k.cast)⟩.fst)) =\n if k = 0 then g ^ minimalPeriod (fun x ↦ g • x) (Quoti... | quotientEquivSigmaZMod_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CoprodI | {
"line": 1057,
"column": 2
} | {
"line": 1057,
"column": 25
} | {
"line": 1059,
"column": 0
} | [
{
"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 : ∀... | [] | exact natCast_le_aleph0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Transfer | {
"line": 217,
"column": 38
} | {
"line": 217,
"column": 67
} | {
"line": 217,
"column": 68
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : ↥H →* A\ninst✝ : H.FiniteIndex\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑⟨g₀⁻¹ * g ^ k * g₀, hk⟩ = g ^ k\n⊢ ↑(List.map\n (fun q ... | [
"G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : ↥H →* A\ninst✝ : H.FiniteIndex\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑⟨g₀⁻¹ * g ^ k * g₀, hk⟩ = g ^ k\n⊢ ↑(List.map\n (fun q ↦\n ... | ← Finset.prod_pow_eq_pow_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Transfer | {
"line": 284,
"column": 42
} | {
"line": 284,
"column": 59
} | {
"line": 284,
"column": 59
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\np : ℕ\nP : Sylow p G\nhP : normalizer ↑P ≤ centralizer ↑P\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Finite (Sylow p G)\ninst✝ : (↑P).FiniteIndex\nhf : Function.Injective ⇑((transferSylow P hP).domRestrict ↑P) ∧ ((transferSylow P hP).domRestrict ↑P).range = ⊤\n⊢ (transferSyl... | [
"G : Type u_1\ninst✝³ : Group G\np : ℕ\nP : Sylow p G\nhP : normalizer ↑P ≤ centralizer ↑P\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Finite (Sylow p G)\ninst✝ : (↑P).FiniteIndex\nhf : Function.Injective ⇑((transferSylow P hP).domRestrict ↑P) ∧ map (transferSylow P hP) ↑P = ⊤\n⊢ (transferSylow P hP).ker.IsComplement' ↑... | domRestrict_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Focal | {
"line": 164,
"column": 53
} | {
"line": 164,
"column": 82
} | {
"line": 165,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.FiniteIndex\nx : ↥H\nthis : Fintype (Quotient (MulAction.orbitRel (↥(zpowers ↑x)) (G ⧸ H)))\n⊢ (QuotientGroup.mk' H.focalSubgroupOf).transfer ↑x = ↑x ^ ∑ q, minimalPeriod (fun x_1 ↦ ↑x • x_1) q.out",
"ppTerm": "?m.41",
"assigned": true,
... | [
"G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.FiniteIndex\nx : ↥H\nthis : Fintype (Quotient (MulAction.orbitRel (↥(zpowers ↑x)) (G ⧸ H)))\n⊢ (QuotientGroup.mk' H.focalSubgroupOf).transfer ↑x = ∏ i, ↑x ^ minimalPeriod (fun x_1 ↦ ↑x • x_1) i.out"
] | ← Finset.prod_pow_eq_pow_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeGroup.Orbit | {
"line": 51,
"column": 2
} | {
"line": 55,
"column": 9
} | {
"line": 56,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\nw : α × Bool\ng : FreeGroup α\nh : g ∉ startsWith (w.1, !w.2)\nhC : 0 < g.toWord.length\n⊢ mk [w] * g ∈ startsWith w",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"List.head",
"Eq.mpr",
"HMul.hMul",
"Bool.not... | [
"case neg\nα : Type u_1\ninst✝ : DecidableEq α\nw : α × Bool\ng : FreeGroup α\nh : g ∉ startsWith (w.1, !w.2)\nhC : ¬0 < g.toWord.length\n⊢ mk [w] * g ∈ startsWith w"
] | · simp only [startsWith, Set.mem_ofPred_eq, getElem?_pos, Option.some.injEq,
Prod.eq_iff_fst_eq_snd_eq, not_and, Bool.not_eq_not, toWord_mul, toWord_mk, reduce.cons,
reduce_nil, List.cons_append, List.nil_append, reduce_toWord, hC] at *
rw [show g.toWord = g.toWord.head (by grind) :: g.toWord.tail by gr... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.FreeGroup.Orbit | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 22
} | {
"line": 87,
"column": 4
} | [
{
"pp": "α : Type u_1\nX : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : MulAction (FreeGroup α) X\nx : X\nw : α × Bool\ng : FreeGroup α\nhg : g ∈ startsWith w\nl : List (α × Bool) := g.toWord\nh : ⟨g, hg⟩ = ⟨mk g.toWord, ⋯⟩\na : α × Bool\nhl : [a] = g.toWord\n⊢ x ∈ (⋃ v ∈ {z | z ≠ (w.1, !w.2)}, MulAction.orbit (↑(... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Nilpotent | {
"line": 551,
"column": 2
} | {
"line": 558,
"column": 37
} | {
"line": 560,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nhG : Group.IsNilpotent G\nn : ℕ\n⊢ upperCentralSeries G n = ⊤ ↔ nilpotencyClass G ≤ n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"eq_top_iff",
"congrArg",
"PartialOrder.toPreorder",
"Subgroup.upperCentralS... | [] | classical
constructor
· intro h
rw [nilpotencyClass_def]
exact Nat.find_le h
· intro h
rw [eq_top_iff, ← upperCentralSeries_nilpotencyClass]
exact upperCentralSeries_mono _ h | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.GroupTheory.Nilpotent | {
"line": 551,
"column": 2
} | {
"line": 558,
"column": 37
} | {
"line": 560,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nhG : Group.IsNilpotent G\nn : ℕ\n⊢ upperCentralSeries G n = ⊤ ↔ nilpotencyClass G ≤ n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"eq_top_iff",
"congrArg",
"PartialOrder.toPreorder",
"Subgroup.upperCentralS... | [] | classical
constructor
· intro h
rw [nilpotencyClass_def]
exact Nat.find_le h
· intro h
rw [eq_top_iff, ← upperCentralSeries_nilpotencyClass]
exact upperCentralSeries_mono _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Nilpotent | {
"line": 551,
"column": 2
} | {
"line": 558,
"column": 37
} | {
"line": 560,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nhG : Group.IsNilpotent G\nn : ℕ\n⊢ upperCentralSeries G n = ⊤ ↔ nilpotencyClass G ≤ n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"eq_top_iff",
"congrArg",
"PartialOrder.toPreorder",
"Subgroup.upperCentralS... | [] | classical
constructor
· intro h
rw [nilpotencyClass_def]
exact Nat.find_le h
· intro h
rw [eq_top_iff, ← upperCentralSeries_nilpotencyClass]
exact upperCentralSeries_mono _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Goursat | {
"line": 143,
"column": 2
} | {
"line": 147,
"column": 53
} | {
"line": 148,
"column": 2
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nG' : Subgroup G := map (MonoidHom.fst G H) I\nH' : Subgroup H := map (MonoidHom.snd G H) I\nP : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I\nQ : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I\nI' : Subgroup (↥G' × ↥... | [
"G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nG' : Subgroup G := map (MonoidHom.fst G H) I\nH' : Subgroup H := map (MonoidHom.snd G H) I\nP : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I\nQ : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I\nI' : Subgroup (↥G' × ↥H') := (P.pr... | have hI₂' : Surjective (Prod.snd ∘ I'.subtype) := by
simp only [← MonoidHom.coe_snd, ← MonoidHom.coe_comp, ← MonoidHom.range_eq_top,
MonoidHom.range_comp, Subgroup.range_subtype, I']
simp only [← MonoidHom.range_comp, MonoidHom.range_eq_top]
exact (MonoidHom.snd ..).subgroupMap_surjective I | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 43
} | {
"line": 421,
"column": 4
} | [
{
"pp": "case cover\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nhGX : IsPretransitive G X\nhB : IsBlock G B\na b : X\nhb : b ∈ B\n⊢ ∃! b, (b ∈ range fun g ↦ g • B) ∧ a ∈ b",
"ppTerm": "?cover",
"assigned": true,
"usedConstants": [
"instHSMul",
"Member... | [
"case cover\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nhGX : IsPretransitive G X\nhB : IsBlock G B\nb : X\nhb : b ∈ B\ng : G\n⊢ ∃! b_1, (b_1 ∈ range fun g ↦ g • B) ∧ g • b ∈ b_1"
] | obtain ⟨g, rfl⟩ := exists_smul_eq G b a | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 545,
"column": 4
} | {
"line": 545,
"column": 43
} | {
"line": 546,
"column": 4
} | [
{
"pp": "case mpr\nG : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\nB : Set X\ninst✝ : IsPretransitive G X\nhB : IsBlock G B\na : X\nha : a ∈ B\nx : X\nhx : x ∈ B\n⊢ x ∈ orbit (↥(stabilizer G B)) a",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"instHSMul",
... | [
"case mpr\nG : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\nB : Set X\ninst✝ : IsPretransitive G X\nhB : IsBlock G B\na : X\nha : a ∈ B\nk : G\nhx : k • a ∈ B\n⊢ k • a ∈ orbit (↥(stabilizer G B)) a"
] | obtain ⟨k, rfl⟩ := exists_smul_eq G a x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.IndexNormal | {
"line": 60,
"column": 2
} | {
"line": 62,
"column": 60
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhHp : H.index = (Nat.card G).minFac\nhG0 : ¬Nat.card G = 0\nhG1 : ¬Nat.card G = 1\nthis : Finite G\nindex_ne_zero : H.index ≠ 0\nhp : Nat.Prime H.index\n⊢ H.normalCore.index = H.index",
"ppTerm": "?neg✝",
"assigned": true,
"usedConsta... | [
"case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhHp : H.index = (Nat.card G).minFac\nhG0 : ¬Nat.card G = 0\nhG1 : ¬Nat.card G = 1\nthis : Finite G\nindex_ne_zero : H.index ≠ 0\nhp : Nat.Prime H.index\nh : H.normalCore.index ∣ H.index !\n⊢ H.normalCore.index = H.index"
] | have h : H.normalCore.index ∣ H.index ! := by
rw [normalCore_eq_ker, index_ker, index_eq_card, ← Nat.card_perm]
exact card_subgroup_dvd_card (toPermHom G (G ⧸ H)).range | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 716,
"column": 4
} | {
"line": 716,
"column": 36
} | {
"line": 717,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\ninst✝ : IsPretransitive G X\nB : Set X\na : X\nhfB : B.Finite\nB' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B\nhfB_ne : B.Nonempty\nhB'₀ : ∀ (k : G), a ∈ k • B → B' ⊆ k • B\n⊢ B'.Finite",
"ppTerm": "?m.83",
"assigned": true,
... | [
"G : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\ninst✝ : IsPretransitive G X\nB : Set X\na : X\nhfB : B.Finite\nB' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B\nhB'₀ : ∀ (k : G), a ∈ k • B → B' ⊆ k • B\nb : X\nhb : b ∈ B\n⊢ B'.Finite"
] | obtain ⟨b, hb : b ∈ B⟩ := hfB_ne | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 29
} | {
"line": 78,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\na : α\nt : Set ↥(ofStabilizer G a)\n⊢ a ∉ Subtype.val '' t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SubMulAction.instSetLike",
"False",
"congrArg",
"Membership.mem",
"Subgrou... | [] | rintro ⟨b, hb⟩
exact b.prop (by simp [hb]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 29
} | {
"line": 78,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\na : α\nt : Set ↥(ofStabilizer G a)\n⊢ a ∉ Subtype.val '' t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SubMulAction.instSetLike",
"False",
"congrArg",
"Membership.mem",
"Subgrou... | [] | rintro ⟨b, hb⟩
exact b.prop (by simp [hb]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 357,
"column": 10
} | {
"line": 357,
"column": 12
} | {
"line": 358,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα4 : 4 ≤ Nat.card α\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ center ↥(alternatingGroup α)\na : α\n⊢ a ∉ g.support",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
"Finset",
... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα4 : 4 ≤ Nat.card α\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ center ↥(alternatingGroup α)\na : α\nha : a ∈ g.support\n⊢ False"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 460,
"column": 6
} | {
"line": 460,
"column": 22
} | {
"line": 461,
"column": 4
} | [
{
"pp": "k : ℕ\nhrec :\n ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],\n IsMultiplyPretransitive G α k →\n ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!\nG : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulActio... | [] | exact succ_pos k | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 493,
"column": 14
} | {
"line": 493,
"column": 51
} | {
"line": 495,
"column": 0
} | [
{
"pp": "case e'_2\nk : ℕ\nhrec :\n ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],\n IsMultiplyPretransitive G α k →\n ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!\nG : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹... | [] | { rw [nat_card_ofStabilizer_eq G a] } | Lean.Elab.Tactic.evalTacticSeqBracketed | Lean.Parser.Tactic.tacticSeqBracketed |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 493,
"column": 14
} | {
"line": 493,
"column": 51
} | {
"line": 495,
"column": 0
} | [
{
"pp": "case e'_2\nk : ℕ\nhrec :\n ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],\n IsMultiplyPretransitive G α k →\n ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!\nG : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹... | [] | { rw [nat_card_ofStabilizer_eq G a] } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 493,
"column": 14
} | {
"line": 493,
"column": 51
} | {
"line": 495,
"column": 0
} | [
{
"pp": "case e'_3\nk : ℕ\nhrec :\n ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],\n IsMultiplyPretransitive G α k →\n ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!\nG : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹... | [] | { rw [nat_card_ofStabilizer_eq G a] } | Lean.Elab.Tactic.evalTacticSeqBracketed | Lean.Parser.Tactic.tacticSeqBracketed |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 493,
"column": 14
} | {
"line": 493,
"column": 51
} | {
"line": 495,
"column": 0
} | [
{
"pp": "case e'_3\nk : ℕ\nhrec :\n ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],\n IsMultiplyPretransitive G α k →\n ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!\nG : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹... | [] | { rw [nat_card_ofStabilizer_eq G a] } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 505,
"column": 86
} | {
"line": 513,
"column": 58
} | {
"line": 515,
"column": 0
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\ns : Set α\ninst✝ : Finite α\nhs : IsPreprimitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)\ng : M\nha : s ∪ g • s ≠ ⊤\n⊢ IsPreprimitive ↥(fixingSubgroup M (s ∩ g • s)) ↥(ofFixingSubgroup M (s ∩ g • s))",
"ppTerm": "?m.37",
... | [] | by
have := IsPretransitive.isPretransitive_ofFixingSubgroup_inter hs.toIsPretransitive ha
apply IsPreprimitive.of_card_lt (f := ofFixingSubgroup_of_inclusion M Set.inter_subset_left)
rw [show Nat.card (ofFixingSubgroup M (s ∩ g • s)) = (s ∩ g • s)ᶜ.ncard from
Nat.card_coe_set_eq _, Set.ncard_range_of_injectiv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Period | {
"line": 79,
"column": 40
} | {
"line": 79,
"column": 55
} | {
"line": 79,
"column": 56
} | [
{
"pp": "α : Type v\nG : Type u\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nn : ℕ\n⊢ (g⁻¹ ^ n)⁻¹ • a = a ↔ g ^ n • a = a",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"instHSMul",
"DivInvOneMonoid.toInvOneClass",
"congrA... | [
"α : Type v\nG : Type u\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nn : ℕ\n⊢ (g⁻¹ ^ ↑n)⁻¹ • a = a ↔ g ^ n • a = a"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 129,
"column": 10
} | {
"line": 129,
"column": 12
} | {
"line": 129,
"column": 13
} | [
{
"pp": "α : Type u_2\ns : Set α\na : α\n⊢ a ∈ s → ∀ b ∈ s, ∃ g ∈ stabilizer (Perm α) s, g • a = b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership",
"Set"
],
"usedFVars": [
"α",
"s",
"a"
],
"usedGoals... | [
"α : Type u_2\ns : Set α\na : α\nha : a ∈ s\n⊢ ∀ b ∈ s, ∃ g ∈ stabilizer (Perm α) s, g • a = b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 22
} | {
"line": 150,
"column": 2
} | [
{
"pp": "α : Type u_2\ns : Set α\nhs : s.Nonempty\nhsc : sᶜ.Nonempty\n⊢ stabilizer (Perm α) s ≠ ⊤",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Membership.mem",
"DivInvMonoid.toMonoid",
"Subgroup",
"Ne",
"Group.toDivInvMon... | [
"α : Type u_2\ns : Set α\nhsc : sᶜ.Nonempty\na : α\nha : a ∈ s\n⊢ stabilizer (Perm α) s ≠ ⊤"
] | obtain ⟨a, ha⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 22
} | {
"line": 139,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Set α\nhs : s.Nonempty\nhsc : sᶜ.Nontrivial\n⊢ stabilizer (↥(alternatingGroup α)) s ≠ ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Membership.mem",
"Subtype",
"D... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Set α\nhsc : sᶜ.Nontrivial\na : α\nha : a ∈ s\n⊢ stabilizer (↥(alternatingGroup α)) s ≠ ⊤"
] | obtain ⟨a, ha⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 151,
"column": 10
} | {
"line": 151,
"column": 12
} | {
"line": 151,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα : 4 ≤ Nat.card α\nt : Set α\na : α\n⊢ a ∈ t → ∀ b ∈ t, ∃ g ∈ stabilizer (↥(alternatingGroup α)) t, g • a = b",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership",
"Set"
... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα : 4 ≤ Nat.card α\nt : Set α\na : α\nha : a ∈ t\n⊢ ∀ b ∈ t, ∃ g ∈ stabilizer (↥(alternatingGroup α)) t, g • a = b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 208,
"column": 12
} | {
"line": 208,
"column": 14
} | {
"line": 208,
"column": 15
} | [
{
"pp": "case hs\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nG : Subgroup M\nhG : stabilizer M s < G\nmoves : ∀ {s : Set α}, ∀ a ∈ s, ∀ b ∈ s, ∃ g ∈ stabilizer M s, g • a = b\na : α\n⊢ a ∈ s → ∀ b ∈ s, ∃ g, g • a = b",
"ppTerm": "?hs",
"assigned": true,
"usedCons... | [
"case hs\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nG : Subgroup M\nhG : stabilizer M s < G\nmoves : ∀ {s : Set α}, ∀ a ∈ s, ∀ b ∈ s, ∃ g ∈ stabilizer M s, g • a = b\na : α\nha : a ∈ s\n⊢ ∀ b ∈ s, ∃ g, g • a = b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 211,
"column": 12
} | {
"line": 211,
"column": 14
} | {
"line": 211,
"column": 15
} | [
{
"pp": "case hs'\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nG : Subgroup M\nhG : stabilizer M s < G\nmoves : ∀ {s : Set α}, ∀ a ∈ s, ∀ b ∈ s, ∃ g ∈ stabilizer M s, g • a = b\na : α\n⊢ a ∈ sᶜ → ∀ b ∈ sᶜ, ∃ g, g • a = b",
"ppTerm": "?hs'",
"assigned": true,
"used... | [
"case hs'\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nG : Subgroup M\nhG : stabilizer M s < G\nmoves : ∀ {s : Set α}, ∀ a ∈ s, ∀ b ∈ s, ∃ g ∈ stabilizer M s, g • a = b\na : α\nha : a ∈ sᶜ\n⊢ ∀ b ∈ sᶜ, ∃ g, g • a = b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 167,
"column": 34
} | {
"line": 167,
"column": 52
} | {
"line": 167,
"column": 52
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα : 4 ≤ Nat.card α\nt : Set α\na : α\nha : a ∈ t\nb : α\nhb : b ∈ t\nhab : ¬a = b\nht : ¬2 < t.ncard\nthis : t.ncard + tᶜ.ncard = Nat.card α\nc d : α\nhc : c ∈ tᶜ\nhd : d ∈ tᶜ\nhcd : c ≠ d\n⊢ swap a b * swap c d ∈ alternatingGroup α",
"ppTer... | [] | by simp [hab, hcd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 78
} | {
"line": 184,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh4 : 4 < Nat.card α\nG : Subgroup ↥(alternatingGroup α)\nhG' : IsPreprimitive (↥G) α\ns : Set α\nhG : stabilizer (↥(alternatingGroup α)) s ≤ G\ng : Perm α\nhg : g ∈ stabilizer (Perm α) s\nhg3 : g.IsThreeCycle\n⊢ g ∈ Subgroup.map (alternatingGroup... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh4 : 4 < Nat.card α\nG : Subgroup ↥(alternatingGroup α)\nhG' : IsPreprimitive (↥G) α\ns : Set α\nhG : stabilizer (↥(alternatingGroup α)) s ≤ G\ng : Perm α\nhg : g ∈ stabilizer (Perm α) s\nhg3 : g.IsThreeCycle\n⊢ IsPreprimitive (↥(Subgroup.map (alternatingGro... | · use ⟨g, hg3.mem_alternatingGroup⟩
simpa only [SetLike.mem_coe, Subgroup.subtype_apply, and_true] using hG hg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 22
} | {
"line": 102,
"column": 22
} | [
{
"pp": "α : Type u_2\ninst✝³ : DecidableEq α\nG : Type u_3\ninst✝² : AddGroup G\ninst✝¹ : AddAction G α\nn : ℕ\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\ninst✝ : FaithfulVAdd G α\n⊢ FaithfulVAdd G ↑(powersetCard α n)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoi... | [
"α : Type u_2\ninst✝³ : DecidableEq α\nG : Type u_3\ninst✝² : AddGroup G\ninst✝¹ : AddAction G α\nn : ℕ\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\ninst✝ : FaithfulVAdd G α\n⊢ ∀ (g : G), (∀ (a : ↑(powersetCard α n)), g +ᵥ a = a) → g = 0"
] | faithfulVAdd_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 446,
"column": 2
} | {
"line": 446,
"column": 43
} | {
"line": 447,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhG : IsPreprimitive (↥G) α\ng : Perm α\nh3g : g.IsThreeCycle\nhg : g ∈ G\nhα4 : Nat.card α ≥ 4\nn : ℕ\nhn : Nat.card α = n + 4\n⊢ IsMultiplyPreprimitive (↥G) α (Nat.card α - 2)",
"ppTerm": "?inr",
"assigne... | [
"case inr\nα : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhG : IsPreprimitive (↥G) α\ng : Perm α\nh3g : g.IsThreeCycle\nhg : g ∈ G\nhα4 : Nat.card α ≥ 4\nn : ℕ\nhn : Nat.card α = n + 4\n⊢ IsMultiplyPreprimitive (↥G) α (n + 2)"
] | rw [show Nat.card α - 2 = n + 2 by grind] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.IsPerfect | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 61
} | {
"line": 102,
"column": 2
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsPerfect G\nhf : Function.Surjective ⇑f\n⊢ IsPerfect G'",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"Group.IsPerfect.top_iff",
"congrArg... | [
"G : Type u_1\nG' : Type u_2\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsPerfect G\nhf : Function.Surjective ⇑f\n⊢ IsPerfect ↥f.range"
] | rw [← top_iff, ← MonoidHom.range_eq_top_of_surjective f hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Cycle.PossibleTypes | {
"line": 43,
"column": 12
} | {
"line": 43,
"column": 14
} | {
"line": 44,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl\nhc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ ... | [
"α : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl\nhc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Finty... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.Perm.Cycle.PossibleTypes | {
"line": 69,
"column": 17
} | {
"line": 69,
"column": 19
} | {
"line": 69,
"column": 20
} | [
{
"pp": "case h.right.right.hf\nα : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl\nhc'_lt :... | [
"case h.right.right.hf\nα : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl\nhc'_lt : ∀ l ∈ c.ran... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.HNNExtension | {
"line": 542,
"column": 6
} | {
"line": 542,
"column": 44
} | {
"line": 543,
"column": 6
} | [
{
"pp": "case neg.inl\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : NormalWord d\nhcan : ¬Cancels 1 w\n⊢ of ↑(⋯.equiv w.head).1 * (of ↑(⋯.equiv w.head).2 * ((of w.head)⁻¹ * ReducedWord.prod φ w.toReducedWord)) =\n ReducedWord.prod φ w.toReducedWord",
"ppTe... | [
"case neg.inl\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : NormalWord d\nhcan : ¬Cancels 1 w\n⊢ of ↑(⋯.equiv w.head).1 *\n (of ((↑(⋯.equiv w.head).1)⁻¹ * w.head) * ((of w.head)⁻¹ * ReducedWord.prod φ w.toReducedWord)) =\n ReducedWord.prod φ w.toReducedWord"
... | erw [(d.compl 1).equiv_snd_eq_inv_mul] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.GroupTheory.Perm.Cycle.PossibleTypes | {
"line": 118,
"column": 16
} | {
"line": 118,
"column": 18
} | {
"line": 118,
"column": 19
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhc : m.sum ≤ Fintype.card α\nh2c : ∀ a ∈ m, 2 ≤ a\nhc' : m.toList.sum ≤ Fintype.card α\np : List (List α)\nhp_length : List.map List.length p = m.toList\nhp_nodup : ∀ s ∈ p, s.Nodup\nhp_disj : List.Pairwise List.Disjoint p\nhp2 : ... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhc : m.sum ≤ Fintype.card α\nh2c : ∀ a ∈ m, 2 ≤ a\nhc' : m.toList.sum ≤ Fintype.card α\np : List (List α)\nhp_length : List.map List.length p = m.toList\nhp_nodup : ∀ s ∈ p, s.Nodup\nhp_disj : List.Pairwise List.Disjoint p\nhp2 : ∀ x ∈ p, 2 ≤... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.HNNExtension | {
"line": 572,
"column": 4
} | {
"line": 581,
"column": 16
} | {
"line": 583,
"column": 0
} | [
{
"pp": "case cons\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\ng : G\nu : ℤˣ\nw : NormalWord d\nh1 : w.head ∈ d.set u\nh2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B u → u = u'\nih : ReducedWord.prod φ w.toReducedWord • empty = w\n⊢ Reduced... | [] | rw [prod_cons, ← mul_assoc, mul_smul, ih, mul_smul, t_pow_smul_eq_unitsSMul,
of_smul_eq_smul, unitsSMul]
rw [dif_neg (not_cancels_of_cons_hyp u w h2)]
-- Before https://github.com/leanprover/lean4/pull/2644, this was just
-- simp [unitsSMulGroup, (d.compl _).equiv_fst_eq_one_of_mem_of_one_mem (one_mem... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.HNNExtension | {
"line": 572,
"column": 4
} | {
"line": 581,
"column": 16
} | {
"line": 583,
"column": 0
} | [
{
"pp": "case cons\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\ng : G\nu : ℤˣ\nw : NormalWord d\nh1 : w.head ∈ d.set u\nh2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B u → u = u'\nih : ReducedWord.prod φ w.toReducedWord • empty = w\n⊢ Reduced... | [] | rw [prod_cons, ← mul_assoc, mul_smul, ih, mul_smul, t_pow_smul_eq_unitsSMul,
of_smul_eq_smul, unitsSMul]
rw [dif_neg (not_cancels_of_cons_hyp u w h2)]
-- Before https://github.com/leanprover/lean4/pull/2644, this was just
-- simp [unitsSMulGroup, (d.compl _).equiv_fst_eq_one_of_mem_of_one_mem (one_mem... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 181,
"column": 17
} | {
"line": 185,
"column": 8
} | {
"line": 187,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nx✝ : Perm ↥g.cycleFactorsFinset\nhσ : x✝ ∈ {τ | ∀ (c : ↥g.cycleFactorsFinset), #(↑(τ c)).support = #(↑c).support}\n⊢ x✝⁻¹ ∈ {τ | ∀ (c : ↥g.cycleFactorsFinset), #(↑(τ c)).support = #(↑c).support}",
"ppTerm": "?m.52",
"assigned"... | [] | by
simp only [Subtype.forall, Set.mem_ofPred_eq] at hσ ⊢
intro c hc
rw [← hσ _ (by simp)]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.PushoutI | {
"line": 348,
"column": 4
} | {
"line": 348,
"column": 58
} | {
"line": 349,
"column": 2
} | [
{
"pp": "case neg\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nw : Word G\ni : ι\nh✝ : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g\n... | [] | · rw [equiv_one (d.compl i) (one_mem _) (d.one_mem _)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 179,
"column": 26
} | {
"line": 179,
"column": 42
} | {
"line": 179,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh : Subgroup.centralizer {g} ≤ alternatingGroup α\nc : Perm α\nhc : c ∈ g.cycleFactorsFinset\nd : Perm α\nhd : d ∈ g.cycleFactorsFinset\nhm : #c.support = #d.support\nhm' : c ≠ d\nτ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh : Subgroup.centralizer {g} ≤ alternatingGroup α\nc : Perm α\nhc : c ∈ g.cycleFactorsFinset\nd : Perm α\nhd : d ∈ g.cycleFactorsFinset\nhm : #c.support = #d.support\nhm' : c ≠ d\nτ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩\na : g.Basi... | Subtype.coe_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 32
} | {
"line": 118,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : ↥(alternatingGroup α)\nhg : g ∈ {g | (↑g).IsThreeCycle}\n⊢ g ∈ (ofSubtype ↑⟨(↑g).support, ⋯⟩).range",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"MonoidHom.range",
... | [] | rw [mem_range_ofSubtype_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 281,
"column": 2
} | {
"line": 284,
"column": 47
} | {
"line": 285,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh5 : 5 ≤ Nat.card α\n⊢ _root_.commutator ↥(alternatingGroup α) = ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.closure",
"eq_top_iff",
"congrArg",
"Set.ofPred",
"Eq... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh5 : 5 ≤ Nat.card α\n⊢ closure {b | (↑b).IsThreeCycle} = ⊤"
] | suffices closure {b : alternatingGroup α | (b : Perm α).IsThreeCycle} = ⊤ by
rw [eq_top_iff, ← this, Subgroup.closure_le]
intro b hb
exact hb.mem_commutator_alternatingGroup h5 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 204,
"column": 4
} | {
"line": 205,
"column": 40
} | {
"line": 206,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nx✝ : (kleinFour α).Normal\nthis : Nat.card (↥(alternatingGroup α) ⧸ kleinFour α) = 3\ncomm_le : commutator ↥(alternatingGroup α) ≤ kleinFour α\n⊢ commutator ↥(alternatingGroup α) ≠ ⊥",
"ppTerm": "?m.81",
"assigned": ... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nx✝ : (kleinFour α).Normal\nthis : Nat.card (↥(alternatingGroup α) ⧸ kleinFour α) = 3\ncomm_le : commutator ↥(alternatingGroup α) ≤ kleinFour α\n⊢ ¬⊥ = ⊤"
] | rw [ne_eq, commutator_eq_bot_iff_center_eq_top,
center_eq_bot (le_of_eq hα4.symm)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 223,
"column": 6
} | {
"line": 226,
"column": 43
} | {
"line": 227,
"column": 6
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nx✝ : (kleinFour α).Normal\nthis : Nat.card (↥(alternatingGroup α) ⧸ kleinFour α) = 3\ncomm_le : commutator ↥(alternatingGroup α) ≤ kleinFour α\ncomm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥\nk : ↥(alternatingG... | [
"case inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nx✝ : (kleinFour α).Normal\nthis : Nat.card (↥(alternatingGroup α) ⧸ kleinFour α) = 3\ncomm_le : commutator ↥(alternatingGroup α) ≤ kleinFour α\ncomm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥\nk : ↥(alternatingGroup α)\nhk ... | suffices (⊤ : Subgroup (alternatingGroup α)) =
Subgroup.map fc.toMonoidHom (⊤ : Subgroup (alternatingGroup α)) by
rw [this, ← Subgroup.map_commutator]
refine Subgroup.mem_map_of_mem _ hk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 69
} | {
"line": 192,
"column": 4
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝⁴ : Group G\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\ninst✝² : IsCyclic G\nhG : IsPGroup p G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : (Nat.card G).Coprime (Nat.card K)\nhc : Nat.card G = 0\n⊢ (∀ (g : G) (k : K), k • g * g⁻¹ = 1) ∨ ∀ (g : G), ∃ k q,... | [
"case pos\nG : Type u_1\ninst✝⁴ : Group G\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\ninst✝² : IsCyclic G\nhG : IsPGroup p G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : Subsingleton K ∧ Nonempty K\nhc : Nat.card G = 0\n⊢ (∀ (g : G) (k : K), k • g * g⁻¹ = 1) ∨ ∀ (g : G), ∃ k q, k • q * q⁻¹ = g"
... | rw [hc, Nat.coprime_zero_left, Nat.card_eq_one_iff_unique] at hGK | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Basic | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 64
} | {
"line": 344,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α β\nγ : Type u_4\nmγ : MeasurableSpace γ\nf : γ → β\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\nhf : MeasurableEmbedding f\n⊢ IsSFiniteKernel (κ.comapRight hf)",
"ppTerm": "?m.24",
"assigned":... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α β\nγ : Type u_4\nmγ : MeasurableSpace γ\nf : γ → β\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\nhf : MeasurableEmbedding f\n⊢ κ.comapRight hf = Kernel.sum fun n ↦ (κ.seq n).comapRight hf"
] | refine ⟨⟨fun n => comapRight (seq κ n) hf, inferInstance, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Basic | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 46
} | {
"line": 398,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → ℝ≥0∞\n⊢ ∫⁻ (b : β), g b ∂(piecewise hs κ η) a = if a ∈ s then ∫⁻ (b : β), g b ∂κ a else ∫⁻ (b : β), g b ∂η a",
"ppTerm":... | [] | simp_rw [piecewise_apply]; split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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