module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 634,
"column": 2
} | {
"line": 642,
"column": 16
} | [
{
"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\nm n : ℕ∞ω\ninst✝¹ :... | have T := nhdsWithin_mono _ (subset_insert _ _)
((contMDiffWithinAt_iff_contMDiffWithinAt_nhdsWithin hm).1
(contMDiffWithinAt_mpullbackWithin_extChartAt_symm hV hs hx hmn))
have A := (continuousAt_extChartAt (I := I) x).continuousWithinAt.preimage_mem_nhdsWithin'' T rfl
apply (nhdsWithin_le_iff.2 _) A
f... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 634,
"column": 2
} | {
"line": 642,
"column": 16
} | [
{
"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\nm n : ℕ∞ω\ninst✝¹ :... | have T := nhdsWithin_mono _ (subset_insert _ _)
((contMDiffWithinAt_iff_contMDiffWithinAt_nhdsWithin hm).1
(contMDiffWithinAt_mpullbackWithin_extChartAt_symm hV hs hx hmn))
have A := (continuousAt_extChartAt (I := I) x).continuousWithinAt.preimage_mem_nhdsWithin'' T rfl
apply (nhdsWithin_le_iff.2 _) A
f... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Immersion | {
"line": 140,
"column": 50
} | {
"line": 140,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\nE'' : Type u\nF : Type u_5\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nH : Type u_7\ninst✝³ : Topolog... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 52
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range Subtype.val) else range Subtype.val = univ",
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
... | simp only [instIsRCLikeNormedField, ↓reduceDIte] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 52
} | [
{
"pp": "n : ℕ\n⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range Subtype.val) else range Subtype.val = univ",
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real.instLE",... | simp only [instIsRCLikeNormedField, ↓reduceDIte] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nn : WithTop ℕ∞\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✝... | convert! hf.continuousOn_tangentMapWithin hmn uniqueMDiffOn_univ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Geometry.Manifold.Immersion | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 10
} | [
{
"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_7\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_11\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsManifold I n M\ns... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 32
} | [
{
"pp": "case e_a.refine_1\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nIs : IsManifold I 1... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Immersion | {
"line": 635,
"column": 36
} | {
"line": 635,
"column": 44
} | [
{
"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_7\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_11\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsManifold I n M\nx... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.Immersion | {
"line": 635,
"column": 36
} | {
"line": 635,
"column": 44
} | [
{
"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_7\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_11\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsManifold I n M\nx... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.Immersion | {
"line": 635,
"column": 36
} | {
"line": 635,
"column": 44
} | [
{
"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_7\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_11\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsManifold I n M\nx... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Immersion | {
"line": 635,
"column": 63
} | {
"line": 635,
"column": 71
} | [
{
"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_7\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_11\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsManifold I n M\nx... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.Immersion | {
"line": 635,
"column": 63
} | {
"line": 635,
"column": 71
} | [
{
"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_7\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_11\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsManifold I n M\nx... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.Immersion | {
"line": 635,
"column": 63
} | {
"line": 635,
"column": 71
} | [
{
"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_7\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_11\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsManifold I n M\nx... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Instances.Icc | {
"line": 184,
"column": 11
} | {
"line": 184,
"column": 51
} | [
{
"pp": "case neg\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\nx y : ℝ\nh : Fact (x < y)\nf : ↑(Icc x y) → M\nw : ↑(Icc x y)\nhw : ¬MDiffAt f w\n... | mdifferentiableWithinAt_comp_projIcc_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 517,
"column": 6
} | {
"line": 520,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nn : WithTop ℕ∞\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\nE' : Type u_5\nins... | have D' : DifferentiableWithinAt 𝕜 (φ' ∘ Prod.fst) (Set.range (Prod.map I I'))
(I ((chartAt H p.1.proj) p.1.proj), I' ((chartAt H' p.2.proj) p.2.proj)) :=
DifferentiableWithinAt.comp (t := Set.range I) _ (by exact D0')
differentiableWithinAt_fst (by simp [mapsTo_fst_prod]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.Instances.Sphere | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv : E\nhv : ‖v‖ = 1\nx : ↑(sphere 0 1)\nhx : ↑x ≠ v\na : ℝ := ((innerSL ℝ) v) ↑x\ny : ↥(ℝ ∙ v)ᗮ := (ℝ ∙ v)ᗮ.orthogonalProjection ↑x\nsplit : ↑x = a • v + ↑y\nhvy : ⟪v, ↑y⟫_ℝ = 0\npythag : 1 = a ^ 2 + ‖y‖ ^ 2\nha : 0 < 1 - a\n⊢ ... | linear_combination 4 * (a - 1) * pythag | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.Geometry.Manifold.Instances.Sphere | {
"line": 242,
"column": 10
} | {
"line": 242,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv : E\nhv : ‖v‖ = 1\nw : ↥(ℝ ∙ v)ᗮ\nh₁ : (ℝ ∙ v)ᗮ.orthogonalProjection v = 0\nh₂ : ⟪v, ↑w⟫ = 0\nh₃ : ⟪v, v⟫ = 1\n⊢ (2 / (1 - ((‖↑w‖ ^ 2 + 4)⁻¹ • 4 • ⟪v, ↑w⟫ + (‖↑w‖ ^ 2 + 4)⁻¹ • (‖↑w‖ ^ 2 - 4) • ⟪v, v⟫))) • (‖↑w‖ ^ 2 + 4)⁻¹ • 4... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 704,
"column": 7
} | {
"line": 730,
"column": 53
} | [] | mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀
_ = mpullbackWithin I I' f (mlieBracketWithin I' V W t) s' x₀ := by
simp only [mpullbackWithin, hs', mfderivWithin_inter u_mem]
_ = mpullbackWithin I I' f (mlieBracketWithin I' V W t') s' x₀ := by
simp only [mpullbackWithin, ht', mlieBracketWithin_int... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique | {
"line": 197,
"column": 2
} | {
"line": 198,
"column": 25
} | [
{
"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\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : IsManifold I 1 M\nγ γ' : ℝ → M\nv : (x : M) → TangentSpace I x\nt₀ : ℝ\ninst✝ : T2... | apply isPreconnected_Ioo.subset_of_closure_inter_subset (s := Ioo a b) (u := s) _
⟨t₀, ⟨ht₀, ⟨h, ht₀⟩⟩⟩ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Constructions.UnitInterval | {
"line": 71,
"column": 4
} | {
"line": 73,
"column": 68
} | [
{
"pp": "case h\ns : Set ↑I\nhs : MeasurableSet s\n⊢ volume (⇑symmMeasurableEquiv ⁻¹' s) = volume s",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"MeasureTheory.Measure.measurePreserving_sub_left",
"MeasurableEquiv.instEquivLike",
"MeasureTheory.Measure.i... | conv_lhs => rw [coe_symmMeasurableEquiv, volume_apply, image_coe_preimage_symm,
← map_apply (by fun_prop) (measurableSet_Icc.subtype_image hs),
volume.measurePreserving_sub_left 1 |>.map_eq, ← volume_apply] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 32
} | [
{
"pp": "case h'f\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 ... | exact differentiableOn_neg _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 32
} | [
{
"pp": "case h'f\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 ... | exact differentiableOn_neg _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 32
} | [
{
"pp": "case h'f\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 ... | exact differentiableOn_neg _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.Riemannian | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 14
} | [
{
"pp": "EB : Type u_1\ninst✝¹⁶ : NormedAddCommGroup EB\ninst✝¹⁵ : NormedSpace ℝ EB\nHB : Type u_2\ninst✝¹⁴ : TopologicalSpace HB\nIB : ModelWithCorners ℝ EB HB\nB : Type u_3\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : ChartedSpace HB B\nF : Type u_4\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace ℝ F\nE : ... | have hb : MDifferentiableWithinAt IM IB b s x := by
simp only [mdifferentiableWithinAt_totalSpace] at hv
exact hv.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 12
} | [
{
"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\nF : Type u_5\ninst✝⁶ : NormedAddCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 12
} | [
{
"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\nF : Type u_5\ninst✝⁶ : NormedAddCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 396,
"column": 38
} | {
"line": 398,
"column": 74
} | [
{
"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... | by
simpa [localFrame_coeff] using
(e.isLocalFrameOn_localFrame_baseSet I 1 b).coeff_apply_of_notMem hx i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Polygon.Basic | {
"line": 85,
"column": 2
} | {
"line": 87,
"column": 62
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\nn : ℕ\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : AddTorsor V P\ninst✝¹ : NeZero n\ninst✝ : Nontrivial R\npoly : Polygon P n\nh : HasNondegenerateVertices R poly\n⊢ poly.HasNondegenerateEdges",
"usedConstants": [
"Eq.mpr",... | obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)
intro i
simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Polygon.Basic | {
"line": 85,
"column": 2
} | {
"line": 87,
"column": 62
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\nn : ℕ\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : AddTorsor V P\ninst✝¹ : NeZero n\ninst✝ : Nontrivial R\npoly : Polygon P n\nh : HasNondegenerateVertices R poly\n⊢ poly.HasNondegenerateEdges",
"usedConstants": [
"Eq.mpr",... | obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)
intro i
simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Polygon.Basic | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 12
} | [
{
"pp": "case h\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\nt₁ t₂ : Triangle R P\nh : ((fun t ↦ { vertices := t.points }) t₁).vertices = ((fun t ↦ { vertices := t.points }) t₂).vertices\n⊢ ∀ (i : Fin (2 + 1)), t₁.points i = t... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 204,
"column": 11
} | {
"line": 204,
"column": 24
} | [
{
"pp": "B : Type u_1\ninst✝⁷ : TopologicalSpace B\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B) → NormedAddCommGroup (E x)\ninst✝² : (x : B) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\ninst✝ : Ve... | ← le_opNorm₂, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.GroupTheory.ClassEquation | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 27
} | [
{
"pp": "case intro.e_a.convert_7\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nval✝ : Fintype G\n⊢ ∀ {x : ConjClasses G}, Fintype.card ↑x.carrier ≠ 0 → (x ∈ noncenter G ↔ x ∈ (noncenter G).toFinset)",
"usedConstants": [
"instFintypeConjClassesOfDecidableRelIsConj",
"congrArg",
"Finse... | simp [Set.mem_toFinset] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ #{1} + 1 + 1 + 1 = 4",
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Finset",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ #{1} + 1 + 1 + 1 = 4",
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Finset",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ #{1} + 1 + 1 + 1 = 4",
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Finset",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ y ∉ {1}",
"usedConstants": [
"False",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"eq_false",
"congrArg... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ y ∉ {1}",
"usedConstants": [
"False",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"eq_false",
"congrArg... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ y ∉ {1}",
"usedConstants": [
"False",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"eq_false",
"congrArg... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ x ∉ {y, 1}",
"usedConstants": [
"False",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"eq_false",
"congr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ x ∉ {y, 1}",
"usedConstants": [
"False",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"eq_false",
"congr... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 20
} | [
{
"pp": "case hs\nG : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ x ∉ {y, 1}",
"usedConstants": [
"False",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"eq_false",
"congr... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 151,
"column": 6
} | {
"line": 151,
"column": 22
} | [
{
"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... | rw [he] at hx hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 146,
"column": 4
} | {
"line": 154,
"column": 56
} | [
{
"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... | · classical
have univ₂ : {e (x * y), e x, e y, (1 : G₂)} = Finset.univ := by
simpa [map_univ_equiv e, map_insert, he]
using congr(Finset.map e.toEmbedding $(eq_finset_univ hx hy hxy))
rw [← Ne, ← e.injective.ne_iff] at hx hy hxy
rw [he] at hx hy
symm
apply eq_of_mem_inser... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 307,
"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\nheq : i + i = i - i\n⊢ r i = 1",
"usedConstants": [
"InvOneClass.toOne",
"ZMod.commRing",
"DivInvOneMonoid.toInvOneClass",
"sub_self",
"AddGroupWithOne.toAddGroup",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.CommutingProbability | {
"line": 184,
"column": 56
} | {
"line": 199,
"column": 29
} | [
{
"pp": "n : ℕ\n⊢ commProb (Product (reciprocalFactors n)) = 1 / ↑n",
"usedConstants": [
"Nat.cast_mul._simp_1",
"Iff.mpr",
"zero_le",
"Rat.instOfNat",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Mathlib.Tactic.FieldSimp.NF.div_eq_eval₁",
"Eq.mpr",
"GroupWithZero.t... | by
by_cases h0 : n = 0
· rw [h0, reciprocalFactors_zero, commProb_cons, commProb_nil, mul_one, Nat.cast_zero, div_zero]
apply commProb_eq_zero_of_infinite
by_cases h1 : n = 1
· rw [h1, reciprocalFactors_one, commProb_nil, Nat.cast_one, div_one]
rcases Nat.even_or_odd n with h2 | h2
· rw [reciprocalFacto... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.CoprodI | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 12
} | [
{
"pp": "case pos\nι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\ninst✝ : (i : ι) → DecidableEq (M i)\ni : ι\nm : M i\nw : Word M\nh : w.fstIdx ≠ some i\nm' : M i\nw' : Word M\nh' : w'.fstIdx ≠ some i\nhm : m = 1\nhm' : m' = 1\nhe : w = w'\n⊢ { head := m, tail := w, fstIdx_ne := h } = { head :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.CoprodI | {
"line": 386,
"column": 4
} | {
"line": 390,
"column": 26
} | [
{
"pp": "case cons.refine_1\nι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝ : Monoid N\nmotive : Word M → Sort u_4\nempty : motive Word.empty\ncons :\n (i : ι) → (m : M i) → (w : Word M) → (h1 : w.fstIdx ≠ some i) → (h2 : m ≠ 1) → motive w → motive (Word.cons m w h1 h2)\nm... | · rw [List.isChain_cons] at h2
simp only [fstIdx, ne_eq, Option.map_eq_some_iff,
Sigma.exists, exists_and_right, exists_eq_right, not_exists]
intro m' hm'
exact h2.1 _ hm' rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.CoprodI | {
"line": 449,
"column": 31
} | {
"line": 449,
"column": 39
} | [
{
"pp": "case pos\nι : Type u_1\nM : ι → Type u_2\ninst✝² : (i : ι) → Monoid (M i)\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\ni j : ι\nm : M i\ntail : Word M\nih :\n ⟨i, m⟩ ∈\n (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x ↦\n if ij... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.CoprodI | {
"line": 449,
"column": 31
} | {
"line": 449,
"column": 39
} | [
{
"pp": "case neg\nι : Type u_1\nM : ι → Type u_2\ninst✝² : (i : ι) → Monoid (M i)\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\ni j : ι\nm : M i\ntail : Word M\nih :\n ⟨i, m⟩ ∈\n (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x ↦\n if ij... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 188,
"column": 2
} | {
"line": 191,
"column": 5
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.length (w * cs.simple i) ≠ cs.length w",
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Eq.mp",
"MulOne.toMul",
"Nat.instMod",
... | intro eq
have length_mod_two := cs.length_mul_mod_two w (s i)
rw [eq, length_simple] at length_mod_two
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 188,
"column": 2
} | {
"line": 191,
"column": 5
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.length (w * cs.simple i) ≠ cs.length w",
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Eq.mp",
"MulOne.toMul",
"Nat.instMod",
... | intro eq
have length_mod_two := cs.length_mul_mod_two w (s i)
rw [eq, length_simple] at length_mod_two
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 279,
"column": 30
} | {
"line": 290,
"column": 39
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nj : ℕ\n⊢ (cs.leftInvSeq ω).getD j 1 = cs.wordProd (take j ω) * (Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_ad... | by
induction ω generalizing j with
| nil => simp
| cons i ω ih =>
dsimp [leftInvSeq]
rcases j with _ | j'
· simp
· rw [getD_cons_succ]
rw [(by simp : 1 = ⇑(MulAut.conj (s i)) 1)]
rw [getD_map]
rw [ih j']
simp [← mul_assoc, wordProd_cons] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 42
} | [
{
"pp": "case inr\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nj : ℕ\nnhj : ¬j < ω.length\n⊢ (cs.wordProd (drop (j + 1) ω))⁻¹ *\n ((Option.map cs.simple ω[j]?).getD 1 *\n (cs.wordProd (drop (j + 1) ω) *\n ((cs.wordProd (drop (j + 1) ... | rw [getElem?_eq_none_iff.mpr (by lia)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 42
} | [
{
"pp": "case inr\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nj : ℕ\nnhj : ¬j < ω.length\n⊢ cs.wordProd (take j ω) *\n ((Option.map cs.simple ω[j]?).getD 1 *\n ((cs.wordProd (take j ω))⁻¹ *\n (cs.wordProd (take j ω) * ((Option.map c... | rw [getElem?_eq_none_iff.mpr (by lia)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 1024,
"column": 70
} | {
"line": 1024,
"column": 93
} | [
{
"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)... | rw [zpow_add, zpow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 1024,
"column": 70
} | {
"line": 1024,
"column": 93
} | [
{
"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)... | rw [zpow_add, zpow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CoprodI | {
"line": 1024,
"column": 70
} | {
"line": 1024,
"column": 93
} | [
{
"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)... | rw [zpow_add, zpow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 418,
"column": 21
} | {
"line": 418,
"column": 33
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\n⊢ j' < (cs.rightInvSeq ω).length → (cs.rightInvSeq ω)[j]? ≠ (cs.rightInvSeq ω)[j']?",
"usedConstants": [
"Nat",
"LT.lt",
"CoxeterS... | j'_lt_length | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 97
} | [
{
"pp": "B : Type u_1\ninst✝⁷ : TopologicalSpace B\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B) → NormedAddCommGroup (E x)\ninst✝² : (x : B) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\ninst✝ : Ve... | filter_upwards [eventually_norm_symmL_trivializationAt_self_comp_lt F E x one_lt_two] with y hy | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 199,
"column": 64
} | {
"line": 199,
"column": 81
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nΦ : Subgroup (G →* Mˣ)\ng : G\n⊢ g ∈\n {\n toEquiv :=\n { toFun := fun H ↦ OrderDual.toDual (restrictHom H Mˣ).ker,\n invF... | RelIso.coe_fn_mk, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.Transfer | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 88
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nhH : ¬H.index = 0\nthis : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH\nf : Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=\n fun q ↦ ⟨g, ⋯⟩ ^ Function.minimalPeriod (fun x ↦ g • x) q.out\nhf : ∀ (q : Quotient (orbitRel (↥(zpow... | simpa only [f, Finset.prod_pow_eq_pow_sum, index_eq_sum_minimalPeriod H g] using key | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.Transfer | {
"line": 319,
"column": 31
} | {
"line": 319,
"column": 39
} | [
{
"pp": "case neg\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Finite G\nP : Sylow (Nat.card G).minFac G\nhP : IsCyclic ↥↑P\nhn : ¬Nat.card G = 1\nthis : Fact (Nat.Prime (Nat.card G).minFac)\nkey : Nat.card (↥(normalizer ↑↑P) ⧸ (centralizer ↑↑P).subgroupOf (normalizer ↑↑P)) ∣ Nat.card (MulAut ↥↑P)\n⊢ normalizer ↑P ... | ← index, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Focal | {
"line": 213,
"column": 2
} | {
"line": 214,
"column": 83
} | [
{
"pp": "case a\nG : Type u_1\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : (↑P).FiniteIndex\n⊢ _root_.commutator G ⊓ ↑P ≤ (↑P).focalSubgroup",
"usedConstants": [
"Sylow.toSubgroup",
"Subgroup.instNormalSubtypeMemFocalSubgroupOf",
"Monoid.toMulOneClass",
... | · apply le_trans ?_ (ker_transferFocal_inf_eq_focalSubgroup P).le
exact inf_le_inf_right _ (Abelianization.commutator_subset_ker P.transferFocal) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Transfer | {
"line": 312,
"column": 46
} | {
"line": 338,
"column": 81
} | [
{
"pp": "G : Type u_3\ninst✝¹ : Group G\ninst✝ : Finite G\np : ℕ\nhp : (Nat.card G).minFac = p\nP : Sylow p G\nhP : IsCyclic ↥↑P\n⊢ normalizer ↑P ≤ centralizer ↑P",
"usedConstants": [
"Nat.gcd",
"Sylow.isPGroup'",
"Sylow.toSubgroup",
"Subgroup.instFiniteSubtypeMem",
"Iff.mpr",
... | by
subst hp
by_cases hn : Nat.card G = 1
· have := (Nat.card_eq_one_iff_unique.mp hn).1
rw [Subsingleton.elim (normalizer _) (centralizer P)]
have := Fact.mk (Nat.minFac_prime hn)
have key := card_dvd_of_injective _ (QuotientGroup.kerLift_injective P.normalizerMonoidHom)
rw [normalizerMonoidHom_ker, ← i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 91,
"column": 24
} | {
"line": 91,
"column": 32
} | [
{
"pp": "α : 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 = []\n⊢ IsReduced (L₁ ++ (replicate (n + 1) L₂).flatten ++ L₃)",
"usedConstants": [
"List.replicate",
"congrArg",
"Eq.mp",
"List.flatten_rep... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 91,
"column": 24
} | {
"line": 91,
"column": 32
} | [
{
"pp": "α : 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 = []\n⊢ IsReduced (L₁ ++ (replicate (n + 1) L₂).flatten ++ L₃)",
"usedConstants": [
"List.replicate",
"congrArg",
"Eq.mp",
"List.flatten_rep... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 91,
"column": 24
} | {
"line": 91,
"column": 32
} | [
{
"pp": "α : 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 = []\n⊢ IsReduced (L₁ ++ (replicate (n + 1) L₂).flatten ++ L₃)",
"usedConstants": [
"List.replicate",
"congrArg",
"Eq.mp",
"List.flatten_rep... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 18
} | [
{
"pp": "case nil\nα : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\nh : IsReduced []\n⊢ IsCyclicallyReduced (reduceCyclically [])",
"usedConstants": [
"congrArg",
"FreeGroup.IsCyclicallyReduced.nil._simp_2",
"FreeGroup.IsCyclicallyReduced",
"FreeGroup.reduceCyclically",
... | case nil => simp | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 18
} | [
{
"pp": "case nil\nα : Type u\ninst✝ : DecidableEq α\n⊢ conjugator [] ++ reduceCyclically [] ++ invRev (conjugator []) = []",
"usedConstants": [
"congrArg",
"List.append_nil",
"FreeGroup.reduceCyclically",
"instHAppendOfAppend",
"List",
"FreeGroup.invRev",
"FreeGrou... | case nil => simp | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 232,
"column": 6
} | {
"line": 232,
"column": 99
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nheq : (fun a ↦ a ^ n) x = (fun a ↦ a ^ n) y\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bo... | simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 58
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((... | obtain ⟨n, rfl⟩ := Nat.exists_eq_add_one_of_ne_zero hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.Nilpotent | {
"line": 347,
"column": 74
} | {
"line": 347,
"column": 95
} | [
{
"pp": "case h.e'_4\nG : Type u_1\ninst✝ : Group G\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊤\nh0 : H 0 = ⊥\nhH : ∀ (x : G) (n : ℕ), x ∈ H (n + 1) → ∀ (g : G), ⁅x, g⁆ ∈ H n\nx : G\nm : ℕ\nhx : x ∈ H (n - m)\ng : G\nhm : m < n\n⊢ H (n + 1 - (m + 1)) = H (n - m)",
"usedConstants": [
"Eq.mpr",
"cong... | Nat.add_sub_add_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 360,
"column": 74
} | {
"line": 360,
"column": 95
} | [
{
"pp": "case h.e'_4\nG : Type u_1\ninst✝ : Group G\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), ⁅x, g⁆ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : m < n\n⊢ H (n - m) = H (n + 1 - (m + 1))",
"usedConstants": [
"Eq.mpr",
... | Nat.add_sub_add_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 529,
"column": 4
} | {
"line": 529,
"column": 83
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝ : Group G\nhG : Group.IsNilpotent G\nn : ℕ\nhn : upperCentralSeries G n = ⊤\n⊢ ∃ H, IsAscendingCentralSeries H ∧ H n = ⊤",
"usedConstants": [
"Subgroup.upperCentralSeries_isAscendingCentralSeries",
"Subgroup.IsAscendingCentralSeries",
"Subgroup.u... | exact ⟨upperCentralSeries G, upperCentralSeries_isAscendingCentralSeries G, hn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 306,
"column": 11
} | {
"line": 306,
"column": 24
} | [
{
"pp": "B : Type u_1\ninst✝⁷ : TopologicalSpace B\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B) → NormedAddCommGroup (E x)\ninst✝² : (x : B) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\ninst✝ : Ve... | ← le_opNorm₂, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.GroupTheory.IndexNormal | {
"line": 61,
"column": 8
} | {
"line": 61,
"column": 26
} | [
{
"pp": "G : 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 !",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Monoi... | normalCore_eq_ker, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 718,
"column": 30
} | {
"line": 718,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\nX : Type u_2\ninst✝² : MulAction G X\ninst✝¹ : IsPretransitive G X\nB : Set X\ninst✝ : Finite X\nhB : IsBlock G B\nhB' : Nat.card X < 2 * (orbit G B).ncard\nthis : B.ncard < 2\n⊢ B.Subsingleton",
"usedConstants": [
"Preorder.toLT",
"Subtype.finite",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 718,
"column": 30
} | {
"line": 718,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\nX : Type u_2\ninst✝² : MulAction G X\ninst✝¹ : IsPretransitive G X\nB : Set X\ninst✝ : Finite X\nhB : IsBlock G B\nhB' : Nat.card X < 2 * (orbit G B).ncard\nthis : B.ncard < 2\n⊢ B.Subsingleton",
"usedConstants": [
"Preorder.toLT",
"Subtype.finite",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 718,
"column": 30
} | {
"line": 718,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\nX : Type u_2\ninst✝² : MulAction G X\ninst✝¹ : IsPretransitive G X\nB : Set X\ninst✝ : Finite X\nhB : IsBlock G B\nhB' : Nat.card X < 2 * (orbit G B).ncard\nthis : B.ncard < 2\n⊢ B.Subsingleton",
"usedConstants": [
"Preorder.toLT",
"Subtype.finite",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 755,
"column": 4
} | {
"line": 755,
"column": 18
} | [
{
"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'\ng... | exact hag g hg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 347,
"column": 4
} | {
"line": 348,
"column": 12
} | [
{
"pp": "case «2»\ng : Perm (Fin 5)\nh1 : g ≠ 1\nh_1 : g.cycleType.card = 2\nh2 : g.cycleType = Multiset.replicate 2 2\nh✝ : 2 * 2 ≤ card (Fin 5)\nh : 2 ≤ 3\nha : Even 2\nh04 : 0 ≠ 4\nh13 : 1 ≠ 3\n⊢ (swap 0 4).Disjoint (swap 1 3)",
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Finset... | · rw [disjoint_iff_disjoint_support, support_swap h04, support_swap h13]
decide | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 487,
"column": 13
} | {
"line": 490,
"column": 88
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nφ : Perm α ≃* Perm α\n⊢ comap φ.toMonoidHom (alternatingGroup α) = alternatingGroup α",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Subgroup.instUniqueOfSubsingleton",
"MulEquiv.instEquivLike",
"Equiv.Perm.eq_alt... | by
nontriviality α
apply eq_alternatingGroup_of_index_eq_two
rw [index_comap_of_surjective _ (Equiv.surjective _), alternatingGroup.index_eq_two] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 83,
"column": 19
} | {
"line": 83,
"column": 45
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\nH : Type u_3\nβ : Type u_4\ninst✝¹ : Group H\ninst✝ : MulAction H β\nσ : G → H\nf : α →ₑ[σ] β\nι : Type u_5\nhf : Injective ⇑f\nm : G\nx : ι ↪ α\n⊢ { toFun := f.toFun ∘ (m • x).toFun, inj' := ⋯ } = σ m • { toFun := f.toFun ∘ x.toFun,... | by ext; simp [f.map_smul'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 12
} | [
{
"pp": "case mp\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\n⊢ IsMultiplyPreprimitive M α 1 → IsPreprimitive M α",
"usedConstants": [
"MulAction.IsMultiplyPreprimitive",
"instOfNatNat",
"Nat",
"OfNat.ofNat"
]
}
] | intro H1 | Lean.Elab.Tactic.evalIntro | null |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 12
} | [
{
"pp": "case mp\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\n⊢ IsMultiplyPreprimitive M α 1 → IsPreprimitive M α",
"usedConstants": [
"MulAction.IsMultiplyPreprimitive",
"instOfNatNat",
"Nat",
"OfNat.ofNat"
]
}
] | intro H1 | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 155,
"column": 51
} | {
"line": 155,
"column": 59
} | [
{
"pp": "α : Type u_2\ns : Set α\na : α\nha : a ∈ s\nb : α\nhb : b ∉ s\nh : stabilizer (Perm α) s = ⊤\n⊢ swap a b ∈ stabilizer (Perm α) s",
"usedConstants": [
"Equiv.Perm.applyMulAction",
"congrArg",
"Equiv.swap",
"Classical.propDecidable",
"Membership.mem",
"DivInvMonoid... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 12
} | [
{
"pp": "case inr.inr.inr.x\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\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 278,
"column": 55
} | {
"line": 278,
"column": 63
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\ns B : Set α\nG : Subgroup M\ninst✝ : IsPreprimitive ↥(stabilizer (↥G) s) ↑s\nhB : IsBlock (↥G) B\nhB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → B.Subsingleton\nhG : stabilizer M s < G\nhBs : B ⊆ s\nhB' : Subtype.val ⁻¹' B =... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 278,
"column": 55
} | {
"line": 278,
"column": 63
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\ns B : Set α\nG : Subgroup M\ninst✝ : IsPreprimitive ↥(stabilizer (↥G) s) ↑s\nhB : IsBlock (↥G) B\nhB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → B.Subsingleton\nhG : stabilizer M s < G\nhBs : B ⊆ s\nhB' : Subtype.val ⁻¹' B =... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 278,
"column": 55
} | {
"line": 278,
"column": 63
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\ns B : Set α\nG : Subgroup M\ninst✝ : IsPreprimitive ↥(stabilizer (↥G) s) ↑s\nhB : IsBlock (↥G) B\nhB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → B.Subsingleton\nhG : stabilizer M s < G\nhBs : B ⊆ s\nhB' : Subtype.val ⁻¹' B =... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 321,
"column": 4
} | {
"line": 321,
"column": 23
} | [
{
"pp": "α : Type u_1\nK : Type u_2\ninst✝¹ : Group K\ninst✝ : MulAction K α\nhα : Nat.card α = 2\nhK : fixedPoints K α ≠ _root_.Set.univ\nn : ℕ\nthis✝ : Finite α\nthis : Fintype α\nh2 : IsMultiplyPretransitive K α 2\n⊢ IsMultiplyPretransitive K α n",
"usedConstants": [
"instOfNatNat",
"LE.le",
... | by_cases hn : n ≤ 2 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 160,
"column": 34
} | {
"line": 160,
"column": 53
} | [
{
"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\nc : α\nhct : c ∈ t\nhc : ¬c = a ∧ c ∉ {b}\n⊢ swap c a * swap a b ∈ alternatingGroup α",
"usedConstants": [
"MonoidHom.instFunLike",
"NonUnitalComm... | by simp [hab, hc.1] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 45
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\nE : Type u_3\ninst✝ : Group E\nS : GroupExtension N E G\nσ σ' : S.Section\ng : G\n⊢ ∃ n, σ g = S.inl n * σ' g",
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"GroupExtension.Section.mul_inv_mem_range_inl",
"Di... | obtain ⟨n, hn⟩ := mul_inv_mem_range_inl σ σ' g
exact ⟨n, by rw [hn, inv_mul_cancel_right]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 45
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\nE : Type u_3\ninst✝ : Group E\nS : GroupExtension N E G\nσ σ' : S.Section\ng : G\n⊢ ∃ n, σ g = S.inl n * σ' g",
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"GroupExtension.Section.mul_inv_mem_range_inl",
"Di... | obtain ⟨n, hn⟩ := mul_inv_mem_range_inl σ σ' g
exact ⟨n, by rw [hn, inv_mul_cancel_right]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.IndexNSmul | {
"line": 63,
"column": 72
} | {
"line": 63,
"column": 80
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : IsTorsionFree ℤ M\nn : ℕ\nhn : n ≠ 0\nx : M\nhx : x ∈ (DistribSMul.toLinearMap ℤ M n).ker\n⊢ x = 0",
"usedConstants": [
"Submodule",
"False",
"instHSMul",
"eq_false",
"DistribSMul.toLinearMap",
"congrArg",
"Dis... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.IndexNSmul | {
"line": 63,
"column": 72
} | {
"line": 63,
"column": 80
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : IsTorsionFree ℤ M\nn : ℕ\nhn : n ≠ 0\nx : M\nhx : x ∈ (DistribSMul.toLinearMap ℤ M n).ker\n⊢ x = 0",
"usedConstants": [
"Submodule",
"False",
"instHSMul",
"eq_false",
"DistribSMul.toLinearMap",
"congrArg",
"Dis... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.IndexNSmul | {
"line": 63,
"column": 72
} | {
"line": 63,
"column": 80
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : IsTorsionFree ℤ M\nn : ℕ\nhn : n ≠ 0\nx : M\nhx : x ∈ (DistribSMul.toLinearMap ℤ M n).ker\n⊢ x = 0",
"usedConstants": [
"Submodule",
"False",
"instHSMul",
"eq_false",
"DistribSMul.toLinearMap",
"congrArg",
"Dis... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.IndexNSmul | {
"line": 69,
"column": 80
} | {
"line": 69,
"column": 88
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : IsTorsionFree ℤ M\nn : ℕ\nhn : n ≠ 0\nx : M\nhx : x ∈ (nsmulAddMonoidHom n).ker\n⊢ x = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"False",
"instHSMul",
"eq_false",
"congrArg",
"AddCommGroup.toAddCommMonoi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.IndexNSmul | {
"line": 69,
"column": 80
} | {
"line": 69,
"column": 88
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : IsTorsionFree ℤ M\nn : ℕ\nhn : n ≠ 0\nx : M\nhx : x ∈ (nsmulAddMonoidHom n).ker\n⊢ x = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"False",
"instHSMul",
"eq_false",
"congrArg",
"AddCommGroup.toAddCommMonoi... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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