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.MeasureTheory.Measure.Hausdorff | {
"line": 752,
"column": 6
} | {
"line": 752,
"column": 42
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nd : ℝ\nhd : 0 ≤ d\nr : 𝕜\nhr : r ≠ 0\ns : Set E\nthis : ∀ {r : 𝕜} (s : Set E), μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH... | refine Eq.trans_le ?_ (this (r • s)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 971,
"column": 4
} | {
"line": 971,
"column": 88
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝¹⁰ : EMetricSpace X\ninst✝⁹ : EMetricSpace Y\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : BorelSpace X\ninst✝⁶ : MeasurableSpace Y\ninst✝⁵ : BorelSpace Y\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝... | set e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq (by simp) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 981,
"column": 4
} | {
"line": 981,
"column": 88
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝¹⁰ : EMetricSpace X\ninst✝⁹ : EMetricSpace Y\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : BorelSpace X\ninst✝⁶ : MeasurableSpace Y\ninst✝⁵ : BorelSpace Y\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝... | set e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq (by simp) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 441,
"column": 88
} | {
"line": 447,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_4\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\n⊢ ContMDiff (I.prod 𝓘(𝕜, E)) I.tangent n ⇑(tangentBundleModelSpaceHomeomorph I).symm",
"usedC... | by
apply contMDiff_iff.2 ⟨Homeomorph.continuous _, fun x y ↦ ?_⟩
apply contDiffOn_id.congr
simp only [mfld_simps, mem_range, TotalSpace.toProd, Equiv.coe_fn_symm_mk, forall_exists_index,
Prod.forall, Prod.mk.injEq]
rintro a b x rfl
simpa [PartialEquiv.prod] using ⟨rfl, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 348,
"column": 2
} | {
"line": 353,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : IsManifold I 1 M\nx y : M\nhy... | have h'y : extChartAt I x y ∈ (extChartAt I x).target := (extChartAt I x).map_source hy
have Z := ContinuousLinearMap.IsInvertible.of_inverse
(mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm h'y)
(mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt h'y)
have : (extChartAt I x).symm ((extChartAt I x)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 348,
"column": 2
} | {
"line": 353,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : IsManifold I 1 M\nx y : M\nhy... | have h'y : extChartAt I x y ∈ (extChartAt I x).target := (extChartAt I x).map_source hy
have Z := ContinuousLinearMap.IsInvertible.of_inverse
(mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm h'y)
(mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt h'y)
have : (extChartAt I x).symm ((extChartAt I x)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 240,
"column": 26
} | {
"line": 240,
"column": 40
} | [
{
"pp": "E : Type u_5\nH : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace ℝ H\nf : E → H\nx : E\nhf : DifferentiableAt ℝ f x\nu : Set E\nhu : u ∈ 𝓝 x\ns : Set H\nhs : Convex ℝ s\nhs' : IsClosed[PseudoMetricSpace.toUniformSpace.toTopological... | ← closure_Iio, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 271,
"column": 46
} | {
"line": 271,
"column": 65
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : WithTop ℕ∞\ninst✝ : IsManifold I ... | by simp [hex, hex'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 285,
"column": 44
} | {
"line": 285,
"column": 75
} | [
{
"pp": "𝕜✝ : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜✝\nE✝ : Type u_2\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace 𝕜✝ E✝\nH✝ : Type u_3\ninst✝⁹ : TopologicalSpace H✝\nI✝ : ModelWithCorners 𝕜✝ E✝ H✝\nM✝ : Type u_4\ninst✝⁸ : TopologicalSpace M✝\ninst✝⁷ : ChartedSpace H✝ M✝\nn✝ : WithTop ℕ∞\ne... | fderivWithin_of_mem_nhds <| hφx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.Diffeomorph | {
"line": 627,
"column": 2
} | {
"line": 627,
"column": 20
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_5\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_9\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nn : ℕ∞ω\nM' : Type u_13\ninst... | exact Sum.swap_inl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 514,
"column": 54
} | {
"line": 514,
"column": 67
} | [
{
"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... | interior_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 114,
"column": 2
} | {
"line": 116,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedSpace 𝕜 F\ninst✝¹⁰ : TopologicalSpace (TotalSpace F E)\ninst✝⁹ : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝⁸ : NormedAddCommGroup EB\ninst✝⁷ : ... | filter_upwards [(trivializationAt F E x).open_baseSet.mem_nhds
(mem_baseSet_trivializationAt F E x)] with y hy
using congr_arg Prod.snd <| (trivializationAt F E x).zeroSection 𝕜 hy | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 339,
"column": 80
} | {
"line": 353,
"column": 97
} | [
{
"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... | by
refine ⟨continuous_snd.continuousAt, ?_⟩
have :
∀ᶠ y in 𝓝[range (I.prod I')] extChartAt (I.prod I') x x,
(extChartAt I' x.2 ∘ Prod.snd ∘ (extChartAt (I.prod I') x).symm) y = y.2 := by
/- porting note: was
apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)
mfld_se... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 463,
"column": 37
} | {
"line": 466,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\ninst✝²⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁹ : NormedAddCommGroup E\ninst✝¹⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁶ : TopologicalSpace M\ninst✝¹⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝¹⁴ : Normed... | by
rw [← mdifferentiableWithinAt_univ] at *
convert! hf.prodMap hg
exact univ_prod_univ.symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 382,
"column": 4
} | {
"line": 385,
"column": 49
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : Fib... | apply eventually_of_mem (U := e.baseSet)
· exact mem_nhdsWithin_of_mem_nhds <|
(e.open_baseSet.mem_nhds <| mem_baseSet_trivializationAt F E x₀)
· exact fun x hx ↦ (e.linear 𝕜 hx).map_neg .. | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 382,
"column": 4
} | {
"line": 385,
"column": 49
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : Fib... | apply eventually_of_mem (U := e.baseSet)
· exact mem_nhdsWithin_of_mem_nhds <|
(e.open_baseSet.mem_nhds <| mem_baseSet_trivializationAt F E x₀)
· exact fun x hx ↦ (e.linear 𝕜 hx).map_neg .. | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 748,
"column": 2
} | {
"line": 748,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_21\ninst✝¹ : TopologicalS... | simpa [mfderivWithin_univ] using (mfderivWithin_sumInr (uniqueMDiffWithinAt_univ I)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 748,
"column": 2
} | {
"line": 748,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_21\ninst✝¹ : TopologicalS... | simpa [mfderivWithin_univ] using (mfderivWithin_sumInr (uniqueMDiffWithinAt_univ I)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 748,
"column": 2
} | {
"line": 748,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_21\ninst✝¹ : TopologicalS... | simpa [mfderivWithin_univ] using (mfderivWithin_sumInr (uniqueMDiffWithinAt_univ I)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 1010,
"column": 10
} | {
"line": 1010,
"column": 42
} | [
{
"pp": "case h.e'_24.h.e'_5.h\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ns : Set M\nz :... | t.erase_insert_of_ne (by grind), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 52
} | [
{
"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... | let t' := t ∩ g ⁻¹' ((extChartAt I (g x₀)).source) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 205,
"column": 52
} | {
"line": 205,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁹ : TopologicalSpace H\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nH' : Type u_5\ninst✝⁴ : TopologicalS... | simp [← mpullbackWithin_univ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 205,
"column": 52
} | {
"line": 205,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁹ : TopologicalSpace H\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nH' : Type u_5\ninst✝⁴ : TopologicalS... | simp [← mpullbackWithin_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 205,
"column": 52
} | {
"line": 205,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁹ : TopologicalSpace H\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nH' : Type u_5\ninst✝⁴ : TopologicalS... | simp [← mpullbackWithin_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 437,
"column": 75
} | {
"line": 439,
"column": 88
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns : Set M\nx : M\nV W : (x : M) → Tan... | by
simpa [mfderivWithin_const] using
mlieBracketWithin_smul_right (mdifferentiableWithinAt_const (c := c)) (V := V) hW hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 498,
"column": 2
} | {
"line": 498,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns : Set M\nx : M\nV W : (x : M) → Tan... | exact mlieBracketWithin_subset (subset_univ _) hs hV hW | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.IntegralCurve.Transform | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nH : Type u_2\ninst✝² : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\ns : Set ℝ\nhγ : IsMIntegralCurveOn γ v s\ndt t : ... | exact (hasFDerivWithinAt_id _ _).add_const _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.IntegralCurve.Transform | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 46
} | [
{
"pp": "case h.e'_12\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nH : Type u_2\ninst✝² : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\nt₀ dt : ℝ\nhγ : IsMIntegralCurveAt ... | · simp only [sub_neg_eq_add, sub_add_cancel] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.Instances.Sphere | {
"line": 142,
"column": 2
} | {
"line": 156,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv : E\n⊢ HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) 0",
"usedConstants": [
"ContinuousLinearMap.comp",
"HasFDerivAt",
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
... | have h₀ : HasFDerivAt (fun w : E => ‖w‖ ^ 2) (0 : StrongDual ℝ E) 0 := by
convert! (hasStrictFDerivAt_norm_sq (0 : E)).hasFDerivAt
simp only [map_zero, smul_zero]
have h₁ : HasFDerivAt (fun w : E => (‖w‖ ^ 2 + 4)⁻¹) (0 : StrongDual ℝ E) 0 := by
convert! (hasFDerivAt_inv _).comp _ (h₀.add (hasFDerivAt_cons... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.Instances.Sphere | {
"line": 142,
"column": 2
} | {
"line": 156,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv : E\n⊢ HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) 0",
"usedConstants": [
"ContinuousLinearMap.comp",
"HasFDerivAt",
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
... | have h₀ : HasFDerivAt (fun w : E => ‖w‖ ^ 2) (0 : StrongDual ℝ E) 0 := by
convert! (hasStrictFDerivAt_norm_sq (0 : E)).hasFDerivAt
simp only [map_zero, smul_zero]
have h₁ : HasFDerivAt (fun w : E => (‖w‖ ^ 2 + 4)⁻¹) (0 : StrongDual ℝ E) 0 := by
convert! (hasFDerivAt_inv _).comp _ (h₀.add (hasFDerivAt_cons... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Instances.Sphere | {
"line": 192,
"column": 10
} | {
"line": 192,
"column": 26
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv : E\nhv : ‖v‖ = 1\nw : ↥(ℝ ∙ v)ᗮ\nhw : ⟪v, ↑w⟫_ℝ = 0\n⊢ (‖↑w‖ ^ 2 + 4)⁻¹ * (‖↑w‖ ^ 2 - 4) < 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNorm... | inv_mul_lt_iff₀' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IntegralCurve.Basic | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 45
} | [
{
"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 ... | hasDerivWithinAt_iff_hasFDerivWithinAt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 222,
"column": 10
} | {
"line": 222,
"column": 29
} | [
{
"pp": "case st\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\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : (x : M) → ENorm (TangentSpace I x)\ninst✝ : ∀ (x : M), ENormSMulClass ℝ (... | ← image_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.Riemannian.Basic | {
"line": 503,
"column": 52
} | {
"line": 505,
"column": 31
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : RiemannianBundle fun x ↦ TangentSpace I x\ninst✝² : IsManifold I 1 M\ninst✝¹ : IsC... | by
rcases setOf_riemannianEDist_lt_subset_nhds I hs with ⟨c, c_pos, hc⟩
exact ⟨c, mod_cast c_pos, hc⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 373,
"column": 4
} | {
"line": 375,
"column": 22
} | [
{
"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... | intro x hx
convert! (e.basisAt b hx).span_eq.ge
simp [hx, basisAt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 373,
"column": 4
} | {
"line": 375,
"column": 22
} | [
{
"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... | intro x hx
convert! (e.basisAt b hx).span_eq.ge
simp [hx, basisAt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Commutator.Finite | {
"line": 56,
"column": 56
} | {
"line": 58,
"column": 96
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite ↑(commutatorSet G)\ninst✝ : Group.FG G\n⊢ (center G).FiniteIndex",
"usedConstants": [
"commutatorSet",
"Subgroup.closure",
"Subgroup.quotientCenterEmbedding",
"Subgroup.FiniteIndex",
"Finset",
"QuotientGroup.instInh... | by
obtain ⟨S, -, hS⟩ := Group.rank_spec G
exact ⟨mt (Finite.card_eq_zero_of_embedding (quotientCenterEmbedding hS)) Finite.card_pos.ne'⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 24
} | [
{
"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 := Fintype.ofFinite G₁\n_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e\nx... | by_cases hxy : x = y | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 306,
"column": 4
} | {
"line": 307,
"column": 12
} | [
{
"pp": "case r\nn : ℕ\nhodd : Odd n\nhne1 : n ≠ 1\ni : ZMod n\nh : ∀ (g : DihedralGroup n), g * r i = r i * g\n⊢ r i = 1",
"usedConstants": [
"InvOneClass.toOne",
"ZMod.commRing",
"DivInvOneMonoid.toInvOneClass",
"sub_self",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | have heq := sr.inj (h (sr i))
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 306,
"column": 4
} | {
"line": 307,
"column": 12
} | [
{
"pp": "case r\nn : ℕ\nhodd : Odd n\nhne1 : n ≠ 1\ni : ZMod n\nh : ∀ (g : DihedralGroup n), g * r i = r i * g\n⊢ r i = 1",
"usedConstants": [
"InvOneClass.toOne",
"ZMod.commRing",
"DivInvOneMonoid.toInvOneClass",
"sub_self",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | have heq := sr.inj (h (sr i))
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 11
} | [
{
"pp": "B : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\n⊢ Injective simple",
"usedConstants": [
"CoxeterSystem"
]
}
] | intro cs1 | Lean.Elab.Tactic.evalIntro | null |
Mathlib.GroupTheory.CoprodI | {
"line": 674,
"column": 10
} | {
"line": 674,
"column": 23
} | [
{
"pp": "case append\nι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝ : Monoid N\ni j i✝ j✝ k✝ l✝ : ι\n_w₁✝ : NeWord M i✝ j✝\n_hne✝ : j✝ ≠ k✝\n_w₂✝ : NeWord M k✝ l✝\n_w₁_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₁✝.toList\n_w₂_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'... | toList_head?, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CoprodI | {
"line": 843,
"column": 2
} | {
"line": 845,
"column": 30
} | [
{
"pp": "ι : 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 → (f i) h •... | have hnot1 : h * w.head ≠ 1 := by
rw [← div_inv_eq_mul]
exact div_ne_one_of_ne hnh | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 273,
"column": 40
} | {
"line": 273,
"column": 56
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nj : ℕ\nh : j < ω.length\n⊢ (cs.rightInvSeq ω).getD j 1 =\n (cs.wordProd (drop (j + 1) ω))⁻¹ * (Option.map cs.simple ω[j]?).getD 1 * cs.wordProd (drop (j + 1) ω)",
"usedConstants": [
"Eq.mp... | getD_rightInvSeq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 111,
"column": 12
} | {
"line": 111,
"column": 42
} | [
{
"pp": "B : Type u_1\nB' : Type u_2\ne : B ≃ B'\nM : CoxeterMatrix B\nn : ℕ\n⊢ (Matrix.of fun i j ↦ if i = j then 1 else if ↑j + 1 = ↑i ∨ ↑i + 1 = ↑j then 3 else 2).IsSymm",
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Equiv.instEquivLike",
"eq_false",
"co... | by unfold Matrix.IsSymm; aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 130,
"column": 12
} | {
"line": 130,
"column": 42
} | [
{
"pp": "B : Type u_1\nB' : Type u_2\ne : B ≃ B'\nM : CoxeterMatrix B\nn : ℕ\n⊢ (Matrix.of fun i j ↦\n if i = j then 1\n else\n if ↑i = n - 1 ∧ ↑j = n - 2 ∨ ↑j = n - 1 ∧ ↑i = n - 2 then 4\n else if ↑j + 1 = ↑i ∨ ↑i + 1 = ↑j then 3 else 2).IsSymm",
"usedConstants": [
"Eq.mpr",
... | by unfold Matrix.IsSymm; aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 152,
"column": 12
} | {
"line": 152,
"column": 42
} | [
{
"pp": "B : Type u_1\nB' : Type u_2\ne : B ≃ B'\nM : CoxeterMatrix B\nn : ℕ\n⊢ (Matrix.of fun i j ↦\n if i = j then 1\n else\n if ↑i = n - 1 ∧ ↑j = n - 3 ∨ ↑j = n - 1 ∧ ↑i = n - 3 then 3\n else if ↑j + 1 = ↑i ∨ ↑i + 1 = ↑j then 3 else 2).IsSymm",
"usedConstants": [
"Eq.mpr",
... | by unfold Matrix.IsSymm; aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 168,
"column": 12
} | {
"line": 168,
"column": 42
} | [
{
"pp": "B : Type u_1\nB' : Type u_2\ne : B ≃ B'\nM : CoxeterMatrix B\nn m : ℕ\n⊢ (Matrix.of fun i j ↦ if i = j then 1 else m + 2).IsSymm",
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Equiv.instEquivLike",
"of_decide_eq_true",
"congrArg",
"Matrix",
... | by unfold Matrix.IsSymm; aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 400,
"column": 19
} | {
"line": 400,
"column": 56
} | [
{
"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.rightInvSeq ω).prod = (cs.wordProd ω)⁻¹\n⊢ (cs.rightInvSeq (i :: ω)).prod = (cs.wordProd (i :: ω))⁻¹",
"usedConstants": [
"mul_inv_cancel_right",
"CoxeterSyst... | simp [rightInvSeq, ih, wordProd_cons] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 400,
"column": 19
} | {
"line": 400,
"column": 56
} | [
{
"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.rightInvSeq ω).prod = (cs.wordProd ω)⁻¹\n⊢ (cs.rightInvSeq (i :: ω)).prod = (cs.wordProd (i :: ω))⁻¹",
"usedConstants": [
"mul_inv_cancel_right",
"CoxeterSyst... | simp [rightInvSeq, ih, wordProd_cons] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 400,
"column": 19
} | {
"line": 400,
"column": 56
} | [
{
"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.rightInvSeq ω).prod = (cs.wordProd ω)⁻¹\n⊢ (cs.rightInvSeq (i :: ω)).prod = (cs.wordProd (i :: ω))⁻¹",
"usedConstants": [
"mul_inv_cancel_right",
"CoxeterSyst... | simp [rightInvSeq, ih, wordProd_cons] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.DoubleCoset | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 38
} | [
{
"pp": "case h.h.a.mp\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\na b : G\nhb : b ∈ H\n⊢ a⁻¹ * (1 * a * b) ∈ H",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"inv_mul_cance... | rwa [one_mul, inv_mul_cancel_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.DoubleCoset | {
"line": 275,
"column": 41
} | {
"line": 277,
"column": 51
} | [
{
"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\n⊢ ∃ a, x = ↑a * q",
"usedCo... | by
obtain ⟨a, ha⟩ : ∃ a : H, a * x = q := Quotient.eq.mp hx
exact ⟨⟨a⁻¹, by simp⟩, eq_inv_mul_of_mul_eq ha⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 94,
"column": 40
} | {
"line": 94,
"column": 54
} | [
{
"pp": "case refine_1\nα : Type u\nL₁ L₂ L₃ : List (α × Bool)\nh₁ : IsCyclicallyReduced L₂\nh₂ : IsReduced (L₁ ++ L₂ ++ L₃)\nn✝ n : ℕ\nhn : n + 1 ≠ 0\nh : ¬L₂ = []\nh' : (replicate (n + 1) L₂).flatten ≠ []\n⊢ IsReduced (L₁ ++ (L₂ ++ (replicate n L₂).flatten))",
"usedConstants": [
"Eq.mpr",
"Lis... | ← append_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 248,
"column": 10
} | {
"line": 248,
"column": 63
} | [
{
"pp": "case refine_1.cons\nG : Type u\ninst✝³ : Groupoid G\ninst✝² : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\nX : Type u\ninst✝ : Group X\nf : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X\nf' ... | rw [homOfPath, F'.map_comp, comp_as_mul, ih, mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 282,
"column": 95
} | {
"line": 283,
"column": 18
} | [
{
"pp": "G : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (symgen b)) = FreeGroup.of (WeaklyConnectedComponent.mk (symgen a))",
"usedConstants": [
"FreeGroup.of",
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
... | ←
mul_inv_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 608,
"column": 6
} | {
"line": 608,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : H✝.Normal\nH : Subgroup G\nhG : Group.IsNilpotent G\n⊢ Group.IsNilpotent ↥H",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Exists",
"Eq.mp",
"id",
"Subtype",
"Subgroup",
"Bo... | nilpotent_iff_lowerCentralSeries | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 670,
"column": 6
} | {
"line": 670,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\ninst✝¹ : Group H\nf : G →* H\nhf1 : f.ker ≤ center G\ninst✝ : Group.IsNilpotent H\n⊢ Group.IsNilpotent G",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Exists",
"id",
"Subgroup",
"Bot.bot",
"Subgroup.lowerCentralSe... | nilpotent_iff_lowerCentralSeries | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 912,
"column": 4
} | {
"line": 912,
"column": 44
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\na b : ℕ\nab : a < b\nhn : upperCentralSeries G a = upperCentralSeries G b\nhG : ¬Group.IsNilpotent G\n⊢ nilpotencyClass G ≤ a",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"LE.le",
"Group.nilpotencyClass... | rw [nilpotencyClass_of_not_nilpotent hG] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Nilpotent | {
"line": 963,
"column": 6
} | {
"line": 963,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\nH : Subgroup G\ninst✝⁴ : H.Normal\nG₁ : Type u_2\nG₂ : Type u_3\ninst✝³ : Group G₁\ninst✝² : Group G₂\ninst✝¹ : IsNilpotent G₁\ninst✝ : IsNilpotent G₂\n⊢ IsNilpotent (G₁ × G₂)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Exists",
"id",
"S... | nilpotent_iff_lowerCentralSeries | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 989,
"column": 46
} | {
"line": 999,
"column": 61
} | [
{
"pp": "η : Type u_2\nGs : η → Type u_3\ninst✝ : (i : η) → Group (Gs i)\nn : ℕ\n⊢ lowerCentralSeries ((i : η) → Gs i) n ≤ pi Set.univ fun i ↦ lowerCentralSeries (Gs i) n",
"usedConstants": [
"le_refl",
"Nat.recAux",
"Trans.trans",
"Subgroup.commutator_mono",
"instReflLe",
... | by
let pi := fun f : ∀ i, Subgroup (Gs i) => Subgroup.pi Set.univ f
induction n with
| zero => simp [pi_top]
| succ n ih =>
calc
lowerCentralSeries (∀ i, Gs i) n.succ = ⁅lowerCentralSeries (∀ i, Gs i) n, ⊤⁆ := rfl
_ ≤ ⁅pi fun i => lowerCentralSeries (Gs i) n, ⊤⁆ := commutator_mono ih (le_refl _)... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Nilpotent | {
"line": 1006,
"column": 6
} | {
"line": 1006,
"column": 38
} | [
{
"pp": "η : Type u_2\nGs : η → Type u_3\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : ∀ (i : η), IsNilpotent (Gs i)\nn : ℕ\nh : ∀ (i : η), nilpotencyClass (Gs i) ≤ n\n⊢ IsNilpotent ((i : η) → Gs i)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Exists",
"id",
"Pi.group",
"Subgr... | nilpotent_iff_lowerCentralSeries | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 1040,
"column": 6
} | {
"line": 1040,
"column": 38
} | [
{
"pp": "case intro\nG : Type u_1\ninst✝⁴ : Group G\nH : Subgroup G\ninst✝³ : H.Normal\nη : Type u_2\nGs : η → Type u_3\ninst✝² : (i : η) → Group (Gs i)\ninst✝¹ : Finite η\ninst✝ : ∀ (i : η), IsNilpotent (Gs i)\nval✝ : Fintype η\n⊢ IsNilpotent ((i : η) → Gs i)",
"usedConstants": [
"Eq.mpr",
"con... | nilpotent_iff_lowerCentralSeries | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Goursat | {
"line": 173,
"column": 4
} | {
"line": 174,
"column": 79
} | [
{
"pp": "case h.e'_3.h\nG : 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' : Su... | simp_rw [G', H', MonoidHom.mem_ker, MonoidHom.coe_prodMap, Prod.map_apply, Subgroup.mem_prod,
Prod.one_eq_mk, Prod.ext_iff, ← MonoidHom.mem_ker, QuotientGroup.ker_mk'] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 34
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\nX : Type u_2\ninst✝² : MulAction G X\nB : Set X\nH : Type u_3\nY : Type u_4\ninst✝¹ : Group H\ninst✝ : MulAction H Y\nφ : H → G\nj : Y →ₑ[φ] X\nhB : IsBlock G B\ng₁ g₂ : H\nhg : g₁ • ⇑j ⁻¹' B ≠ g₂ • ⇑j ⁻¹' B\n⊢ Disjoint (g₁ • ⇑j ⁻¹' B) (g₂ • ⇑j ⁻¹' B)",
"usedConstant... | ← Group.preimage_smul_setₛₗ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.ConjAct | {
"line": 44,
"column": 2
} | {
"line": 45,
"column": 88
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nk : ConjAct (Perm α)\ng : Perm α\n⊢ (k • g).support = ConjAct.ofConjAct k • g.support",
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Eq.mpr",
"Finset.inv_smul_mem_iff",
"Equiv.Perm.support",
"Equiv.Perm.smul_... | ext
rw [mem_conj_support, ← Perm.smul_def, ConjAct.ofConjAct_inv, Finset.inv_smul_mem_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.ConjAct | {
"line": 44,
"column": 2
} | {
"line": 45,
"column": 88
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nk : ConjAct (Perm α)\ng : Perm α\n⊢ (k • g).support = ConjAct.ofConjAct k • g.support",
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Eq.mpr",
"Finset.inv_smul_mem_iff",
"Equiv.Perm.support",
"Equiv.Perm.smul_... | ext
rw [mem_conj_support, ← Perm.smul_def, ConjAct.ofConjAct_inv, Finset.inv_smul_mem_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 405,
"column": 41
} | {
"line": 405,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nH : Subgroup G\nhB : IsBlock (↥H) B\ng : G\nh' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)\nh : G\nhH : h ∈ H\nhh : g * h * g⁻¹ = ↑h'\n⊢ (g * h * g⁻¹ * g) • B = g • h • B",
"usedConstants": [
"Eq.mp... | smul_smul g h B, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 764,
"column": 4
} | {
"line": 764,
"column": 13
} | [
{
"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\nhfB' : B'.Finite\nhag : ∀ (g : G), a ∈ g • B' → B' ≤ g • B'\nh... | exact hb' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 304,
"column": 70
} | {
"line": 304,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh2 : Nat.card α ≤ 2\na✝ : Nontrivial α\nhα' : Nat.card α = 2\n⊢ Nat.factorial 2 = 2",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Nat.factorial_two",
"Nat.factorial",
"Nat",
... | Nat.factorial_two | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 47
} | [
{
"pp": "case inr.left\nM : Type u_1\nα : Type u_2\ninst✝³ : Group M\ninst✝² : MulAction M α\ninst✝¹ : IsPretransitive M α\nn : ℕ\na : α\ninst✝ : IsMultiplyPreprimitive M α n.succ\nh1 : n ≥ 1\n⊢ IsMultiplyPretransitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n",
"usedConstants": [
"SubMulAction.mulA... | rw [← ofStabilizer.isMultiplyPretransitive] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 476,
"column": 4
} | {
"line": 485,
"column": 46
} | [
{
"pp": "case succ\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... | have htcard : t.ncard = k := by
rw [← Nat.succ_inj, Nat.succ_eq_add_one, Nat.succ_eq_add_one, ← hs, hat', eq_comm]
suffices ¬ a ∈ (Subtype.val '' t) by
convert! Set.ncard_insert_of_notMem this ?_
· rw [Set.ncard_image_of_injective _ Subtype.coe_injective]
apply Set.toFinite
int... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.GroupAction.OfQuotient | {
"line": 31,
"column": 19
} | {
"line": 31,
"column": 51
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nA : Type u_2\ninst✝¹ : MulAction G A\nH : Subgroup G\ninst✝ : H.Normal\ng : G\nhg : g ∈ H\n⊢ g ∈ toEndHom.ker",
"usedConstants": [
"MulOne.toOne",
"MonoidHom.instFunLike",
"Function.End",
"MonoidHom",
"Monoid.toMulOneClass",
"Membe... | funext a; ext; exact a.2 ⟨g, hg⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.OfQuotient | {
"line": 31,
"column": 19
} | {
"line": 31,
"column": 51
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nA : Type u_2\ninst✝¹ : MulAction G A\nH : Subgroup G\ninst✝ : H.Normal\ng : G\nhg : g ∈ H\n⊢ g ∈ toEndHom.ker",
"usedConstants": [
"MulOne.toOne",
"MonoidHom.instFunLike",
"Function.End",
"MonoidHom",
"Monoid.toMulOneClass",
"Membe... | funext a; ext; exact a.2 ⟨g, hg⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 195,
"column": 27
} | {
"line": 195,
"column": 44
} | [
{
"pp": "case out\nα : Type u_2\ninst✝ : DecidableEq α\ns : Set α\nG : Subgroup (Perm α)\nhG : stabilizer (Perm α) s < G\nthis : ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t\nh1' : s.encard ≤ 1\nh1c' : sᶜ.encard ≤ 1\nhα✝ : univ.encard = 2\nx✝ : Finite α\nhα : Nat.card α = 2\n⊢ Nat.Pri... | Nat.factorial_two | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 17
} | [
{
"pp": "case hM\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\n⊢ stabilizer (↥G) s ≠ ⊤",
"usedConstants": [
"Preorder.toLT",
"Mathlib.Tactic.Contr... | contrapose hG | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 266,
"column": 8
} | {
"line": 266,
"column": 38
} | [
{
"pp": "n : ℕ\nhrec :\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = n + 1 →\n n + 2 < Nat.card α →\n IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → Finite α → IsMultiplyPreprim... | rw [← is_one_preprimitive_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 415,
"column": 4
} | {
"line": 417,
"column": 21
} | [
{
"pp": "case inr.inr\nα : Type u_2\ninst✝ : Finite α\ns : Set α\nhs_nonempty : s.Nonempty\nhsc_nonempty : sᶜ.Nonempty\nhα : Nat.card α ≠ 2 * s.ncard\nh : sᶜ.ncard < s.ncard\n⊢ IsCoatom (stabilizer (Perm α) s)",
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Eq.mpr",
"compl_compl",
... | rw [← stabilizer_compl]
apply isCoatom_stabilizer_of_ncard_lt_ncard_compl hsc_nonempty
rwa [compl_compl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 415,
"column": 4
} | {
"line": 417,
"column": 21
} | [
{
"pp": "case inr.inr\nα : Type u_2\ninst✝ : Finite α\ns : Set α\nhs_nonempty : s.Nonempty\nhsc_nonempty : sᶜ.Nonempty\nhα : Nat.card α ≠ 2 * s.ncard\nh : sᶜ.ncard < s.ncard\n⊢ IsCoatom (stabilizer (Perm α) s)",
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Eq.mpr",
"compl_compl",
... | rw [← stabilizer_compl]
apply isCoatom_stabilizer_of_ncard_lt_ncard_compl hsc_nonempty
rwa [compl_compl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 398,
"column": 8
} | {
"line": 398,
"column": 25
} | [
{
"pp": "case inl.h\nα : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : DecidableEq α\ninst✝ : Finite α\nhG : IsPreprimitive (↥G) α\ng : Perm α\nh2g : g.IsSwap\nhg : g ∈ G\nthis : Fintype α\nhα3 : Nat.card α ≤ 2\n⊢ Nat.factorial 2 ≤ Fintype.card ↥G",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Cl... | Nat.factorial_two | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 38
} | [
{
"pp": "case property\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Set α\nhs : sᶜ.Nontrivial\nk : Perm α\nhk_swap : k.IsSwap\nhk_support : _root_.Disjoint s ↑k.support\nhks : k • s = s\ng : Perm ↑s\nhsg : sign g = 1\n⊢ ⟨Perm.ofSubtype g, ⋯⟩ ∈ stabilizer (↥(alternatingGroup α)) s",
"usedCon... | rw [mem_stabilizer_iff, Submonoid.mk_smul]
exact ofSubtype_mem_stabilizer g | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 38
} | [
{
"pp": "case property\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Set α\nhs : sᶜ.Nontrivial\nk : Perm α\nhk_swap : k.IsSwap\nhk_support : _root_.Disjoint s ↑k.support\nhks : k • s = s\ng : Perm ↑s\nhsg : sign g = 1\n⊢ ⟨Perm.ofSubtype g, ⋯⟩ ∈ stabilizer (↥(alternatingGroup α)) s",
"usedCon... | rw [mem_stabilizer_iff, Submonoid.mk_smul]
exact ofSubtype_mem_stabilizer g | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 42
} | [
{
"pp": "case h.H.a\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Set α\nhs : sᶜ.Nontrivial\nk : Perm α\nhk_swap : k.IsSwap\nhk_support : _root_.Disjoint s ↑k.support\nhks : k • s = s\ng : Perm ↑s\nhsg : sign g = -1\nx : ↑s\n⊢ k ↑x = ↑x",
"usedConstants": [
"Equiv.Perm.support",
... | rw [Set.disjoint_left] at hk_support | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.DomMulAct | {
"line": 113,
"column": 2
} | {
"line": 116,
"column": 25
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nf : α → ι\ninst✝¹ : Finite α\ninst✝ : Fintype ι\n⊢ {g | f ∘ ⇑g = f}.ncard = ∏ i, {a | f a = i}.ncard !",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Finset.univ",
"congrArg",
"Finset",
"setOf",
"Classical.propDecidable",... | classical
cases nonempty_fintype α
simp only [← Nat.card_coe_set_eq, Set.coe_setOf, card_eq_fintype_card]
exact stabilizer_card f | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.GroupTheory.Perm.DomMulAct | {
"line": 113,
"column": 2
} | {
"line": 116,
"column": 25
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nf : α → ι\ninst✝¹ : Finite α\ninst✝ : Fintype ι\n⊢ {g | f ∘ ⇑g = f}.ncard = ∏ i, {a | f a = i}.ncard !",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Finset.univ",
"congrArg",
"Finset",
"setOf",
"Classical.propDecidable",... | classical
cases nonempty_fintype α
simp only [← Nat.card_coe_set_eq, Set.coe_setOf, card_eq_fintype_card]
exact stabilizer_card f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.DomMulAct | {
"line": 113,
"column": 2
} | {
"line": 116,
"column": 25
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nf : α → ι\ninst✝¹ : Finite α\ninst✝ : Fintype ι\n⊢ {g | f ∘ ⇑g = f}.ncard = ∏ i, {a | f a = i}.ncard !",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Finset.univ",
"congrArg",
"Finset",
"setOf",
"Classical.propDecidable",... | classical
cases nonempty_fintype α
simp only [← Nat.card_coe_set_eq, Set.coe_setOf, card_eq_fintype_card]
exact stabilizer_card f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.HNNExtension | {
"line": 534,
"column": 6
} | {
"line": 537,
"column": 22
} | [
{
"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 ↑((toSubgroupEquiv φ 1) (⋯.equiv w.head).1) *\n (t ^ ↑1 * (of ↑(⋯.equiv w.head).2 * (of w.head)⁻¹ * ReducedWord.prod φ w.toReducedWord)) =\n t ^ ↑1... | simp only [toSubgroup_neg_one, toSubgroup_one, toSubgroupEquiv_one, equiv_eq_conj, mul_assoc,
Units.val_one, zpow_one, inv_mul_cancel_left, mul_right_inj]
erw [(d.compl 1).equiv_snd_eq_inv_mul]
simp [mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.HNNExtension | {
"line": 534,
"column": 6
} | {
"line": 537,
"column": 22
} | [
{
"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 ↑((toSubgroupEquiv φ 1) (⋯.equiv w.head).1) *\n (t ^ ↑1 * (of ↑(⋯.equiv w.head).2 * (of w.head)⁻¹ * ReducedWord.prod φ w.toReducedWord)) =\n t ^ ↑1... | simp only [toSubgroup_neg_one, toSubgroup_one, toSubgroupEquiv_one, equiv_eq_conj, mul_assoc,
Units.val_one, zpow_one, inv_mul_cancel_left, mul_right_inj]
erw [(d.compl 1).equiv_snd_eq_inv_mul]
simp [mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 443,
"column": 8
} | {
"line": 443,
"column": 22
} | [
{
"pp": "case mp\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nk : ↥(centralizer {g})\nc : ↥g.cycleFactorsFinset\n⊢ #(↑(((toPermHom g) k) c)).support = #(↑c).support",
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"MonoidHom.instFunLike",
"Equiv.instEquiv... | coe_toPermHom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 49
} | [
{
"pp": "case e_α.e_p.h.a\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nsc : ↥g.cycleFactorsFinset → ℕ := ⋯\nhsc : sc = fun c ↦ #(↑c).support\nx✝ : Perm ↥g.cycleFactorsFinset\n⊢ x✝ ∈ (toPermHom g).range ↔ x✝ ∈ {k | sc ∘ ⇑k = sc}",
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.... | rw [mem_range_toPermHom_iff', Set.mem_setOf_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.HNNExtension | {
"line": 637,
"column": 4
} | {
"line": 638,
"column": 29
} | [
{
"pp": "case cons.refine_1\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : ReducedWord G A B\na : ℤˣ × G\nl : List (ℤˣ × G)\nchain : List.IsChain (fun a b ↦ a.2 ∈ toSubgroup A B a.1 → a.1 = b.1) (a :: l)\nw' : NormalWord d\nhw'1 : ReducedWord.prod φ w'.toReducedWo... | · rw [prod_smul, hw'1]
simp [ReducedWord.prod] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 57,
"column": 21
} | {
"line": 57,
"column": 83
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsMulCommutative ↥H\ninst✝ : H.FiniteIndex\nα β : H.LeftTransversal\nhH : H.Normal\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex\n⊢ ⟨(unop g)⁻¹ * ↑1 * unop g, ⋯⟩ = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne... | rw [Subtype.ext_iff, coe_mk, coe_one, mul_one, inv_mul_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 57,
"column": 21
} | {
"line": 57,
"column": 83
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsMulCommutative ↥H\ninst✝ : H.FiniteIndex\nα β : H.LeftTransversal\nhH : H.Normal\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex\n⊢ ⟨(unop g)⁻¹ * ↑1 * unop g, ⋯⟩ = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne... | rw [Subtype.ext_iff, coe_mk, coe_one, mul_one, inv_mul_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 57,
"column": 21
} | {
"line": 57,
"column": 83
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsMulCommutative ↥H\ninst✝ : H.FiniteIndex\nα β : H.LeftTransversal\nhH : H.Normal\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex\n⊢ ⟨(unop g)⁻¹ * ↑1 * unop g, ⋯⟩ = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne... | rw [Subtype.ext_iff, coe_mk, coe_one, mul_one, inv_mul_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 89,
"column": 30
} | {
"line": 89,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\ng : Perm α\nn : ℕ\nhg : orderOf g ∣ 2 ^ n\nk : ℕ\nhk : 4 ∈ g.cycleType\nhk4 : 4 ≤ 4\nhk1 : 1 < 4\nhg0 : 4 ≠ 2\nt : Multiset ℕ\nh1 : t = Multiset.replicate t.card 0\nht : g.cycleType = 4 ::ₘ t\nh : 0 ∉ g.cycleType\n⊢ t = 0",
... | simpa [h1 ▸ ht, Multiset.mem_replicate] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 210,
"column": 50
} | {
"line": 210,
"column": 59
} | [
{
"pp": "case refine_1\nG : Type u\ninst✝³ : Group G\nN : Subgroup G\ninst✝² : N.Normal\nh1 : (Nat.card ↥N).Coprime N.index\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [Finite G'],\n Nat.card G' < Nat.card G →\n ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 218,
"column": 71
} | {
"line": 221,
"column": 22
} | [
{
"pp": "⊢ IsCyclic (QuaternionGroup 1)",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"QuaternionGroup.card",
"Nat.instMulOneClass",
"Fintype.card",
"QuaternionGroup.instFintypeO... | by
apply isCyclic_of_orderOf_eq_card
· rw [Nat.card_eq_fintype_card, card, mul_one]
exact orderOf_xa 0 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 61
} | [
{
"pp": "case inl\n⊢ Monoid.exponent (QuaternionGroup 0) = 0",
"usedConstants": [
"QuaternionGroup.orderOf_a_one",
"Nat.instMulZeroClass",
"HMul.hMul",
"ZMod.commRing",
"AddGroupWithOne.toAddMonoidWithOne",
"DivInvMonoid.toMonoid",
"instMulNat",
"instOfNatNat"... | exact Monoid.exponent_eq_zero_of_order_zero orderOf_a_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 50
} | [
{
"pp": "case inr.a.a.a\nn : ℕ\nhn : NeZero n\nm : ZMod (2 * n)\n⊢ a m ^ lcm (2 * n) 4 = 1",
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"instNormalizedGCDMonoidNat",
"MulOne.toOne",
"Dvd.dvd",
"instHDiv",
"HMul.hMul",
"MulZeroClass.toMul",
"Monoid.toMulOneC... | rw [← orderOf_dvd_iff_pow_eq_one, orderOf_a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.InformationTheory.Coding.UniquelyDecodable | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 45
} | [
{
"pp": "case a\nα : Type u_1\nS : Set (List α)\nh : UniquelyDecodable S\nL₁ L₂ : { L // ∀ (x : List α), x ∈ L → x ∈ S }\nhflat : (fun L ↦ (↑L).flatten) L₁ = (fun L ↦ (↑L).flatten) L₂\n⊢ ↑L₁ = ↑L₂",
"usedConstants": [
"Membership.mem",
"List",
"List.instMembership",
"Subtype.prop",
... | exact h L₁.val L₂.val L₁.prop L₂.prop hflat | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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