module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 137,
"column": 42
} | {
"line": 137,
"column": 52
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : MeasurableSpace X\ninst✝³ : BorelSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nf : X → Y\nd : ℕ\nhf : Isometry f\ns : Set X\n⊢ (volume.addHaarScalarFactor μH[↑d] • μH[↑d]) (f '' s) = (volume.addHaarScala... | smul_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 142,
"column": 42
} | {
"line": 142,
"column": 52
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : MeasurableSpace X\ninst✝³ : BorelSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nf : X → Y\nd : ℕ\nhf : Isometry f\ns : Set Y\n⊢ (volume.addHaarScalarFactor μH[↑d] • μH[↑d]) (f ⁻¹' s) =\n (volume.addHaa... | smul_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 170,
"column": 42
} | {
"line": 170,
"column": 52
} | [
{
"pp": "𝕜 : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedSpace 𝕜 V\ninst✝³ : MeasurableSpace P\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : BorelSpace P\nd : ℕ\nx : P\nc : 𝕜\nhc : c ≠ 0\ns : Set P\n⊢ (volume.addHaarScalarFacto... | smul_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 176,
"column": 42
} | {
"line": 176,
"column": 52
} | [
{
"pp": "𝕜 : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedSpace 𝕜 V\ninst✝³ : MeasurableSpace P\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : BorelSpace P\nd : ℕ\nx : P\nc : 𝕜\nhc : c ≠ 0\ns : Set P\n⊢ (volume.addHaarScalarFacto... | smul_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 249,
"column": 25
} | {
"line": 249,
"column": 72
} | [
{
"pp": "P : Type u_2\ninst✝ : MetricSpace P\ns : Sphere P\np : P\nhr : 0 ≤ s.radius\n⊢ dist p s.center ^ 2 ≤ s.radius ^ 2 ↔ dist p s.center ≤ s.radius",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Nat.instMulZeroClass",
"Real.partialOrder",
"Real.instLE",
"Re... | pow_le_pow_iff_left₀ dist_nonneg hr two_ne_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 36
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nm : Set X → ℝ≥0∞\ns t : Set X\nr : ℝ≥0∞\nr0 : 0 < r\nhr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y\n⊢ (fun r ↦ (mkMetric'.pre m r) (s ∪ t)) =ᶠ[𝓝[>] 0] fun x ↦ (mkMetric'.pre m x) s + (mkMetric'.pre m x) t",
"usedConstants": [
"Set.Ioi",
"CommSemiring.toSemi... | filter_upwards [Ioo_mem_nhdsGT r0] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 18
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\n⊢ ⊤ ≤ ⨆ r, ⨆ (_ : r > 0), ⊤",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"iSup",
"MeasureTheory.OuterMeasure.instSupSet",
"PartialOrder.toPre... | rw [le_iSup_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 770,
"column": 2
} | {
"line": 797,
"column": 58
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nf : X → Y\nK : ℝ≥0\nd : ℝ\nhf : AntilipschitzWith K f\nhd : 0 ≤ d\ns : Set Y\n⊢ μH[d] (f ⁻¹' s) ≤ ↑K ^ d * μH[d] s",
"us... | rcases eq_or_ne K 0 with (rfl | h0)
· rcases eq_empty_or_nonempty (f ⁻¹' s) with (hs | ⟨x, hx⟩)
· simp only [hs, measure_empty, zero_le]
have : f ⁻¹' s = {x} := by
haveI : Subsingleton X := hf.subsingleton
have : (f ⁻¹' s).Subsingleton := subsingleton_univ.anti (subset_univ _)
exact (subsing... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 770,
"column": 2
} | {
"line": 797,
"column": 58
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nf : X → Y\nK : ℝ≥0\nd : ℝ\nhf : AntilipschitzWith K f\nhd : 0 ≤ d\ns : Set Y\n⊢ μH[d] (f ⁻¹' s) ≤ ↑K ^ d * μH[d] s",
"us... | rcases eq_or_ne K 0 with (rfl | h0)
· rcases eq_empty_or_nonempty (f ⁻¹' s) with (hs | ⟨x, hx⟩)
· simp only [hs, measure_empty, zero_le]
have : f ⁻¹' s = {x} := by
haveI : Subsingleton X := hf.subsingleton
have : (f ⁻¹' s).Subsingleton := subsingleton_univ.anti (subset_univ _)
exact (subsing... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 185,
"column": 25
} | {
"line": 188,
"column": 9
} | [
{
"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\nE' : Type u_5\ninst✝⁹ : NormedA... | by
have : (ContinuousLinearMap.id _ _ : TangentSpace I x →L[𝕜] TangentSpace I x) y = y := rfl
conv_rhs => rw [← this, ← he.symm_comp_deriv hx]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Diffeomorph | {
"line": 417,
"column": 8
} | {
"line": 417,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝¹⁵ : NormedAddCommGroup E'\ninst✝¹⁴ : NormedSpace 𝕜 E'\nF : Type u_4\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\nH : Type u_5\ninst✝¹¹ : T... | ← ContinuousLinearEquiv.image_eq_preimage_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.MFDeriv.Tangent | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 20
} | [
{
"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\np q : Tange... | dsimp [tangentMap] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 49
} | [
{
"pp": "𝕜 : 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 : FiberBundle F E\ni... | rw [mdifferentiableWithinAt_section] at hs ht ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 290,
"column": 51
} | {
"line": 292,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nx : M\nV W V₁ W₁ : (x : M) → TangentSp... | by
rw [← mlieBracketWithin_univ, ← mlieBracketWithin_univ,
hV.mlieBracketWithin_vectorField_eq_nhds hW] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 80
} | [
{
"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... | let B (x₀) : TangentSpace 𝓘(𝕜, E) x₀ := f' x₀ • lieBracketWithin 𝕜 V' W' s' x₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Manifold.Instances.Icc | {
"line": 131,
"column": 2
} | {
"line": 141,
"column": 25
} | [
{
"pp": "case neg\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ℝ\nhz : z ∈ Icc x y\nh'z : ¬↑(projIcc x y ⋯ z) < y\n⊢ ContDiffWithinAt ℝ n ((↑(𝓡∂ 1) ∘ ↑(IccRightChart x y)) ∘ projIcc x y ⋯) (Icc x y) z",
"usedConstants": [
"ContDiff.sub",
"Iff.mpr",
"Eq.mpr",
"InnerProductSpace.to... | · have : ContDiff ℝ n (fun (w : ℝ) ↦
(show EuclideanSpace ℝ (Fin 1) from toLp 2 fun (_ : Fin 1) ↦ y - w)) := by
dsimp
apply contDiff_euclidean.2 (fun i ↦ by fun_prop)
apply this.contDiffWithinAt.congr_of_eventuallyEq_of_mem _ hz
filter_upwards [self_mem_nhdsWithin] with w hw
ext i
su... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.Instances.Sphere | {
"line": 566,
"column": 6
} | {
"line": 566,
"column": 64
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nc : Circle → ℂ := Subtype.val\n⊢ (Continuous fun z ↦ z.1 * z.2) ∧\n ∀ (x : ℂ × ℂ) (y : ℂ),\n ContDiffOn ℝ ω (↑(extChartAt 𝓘(ℝ, ℂ) y) ∘ (fun z ↦ z.1 * z.2) ∘ ↑(extChartAt (𝓘(ℝ, ℂ).prod 𝓘(ℝ, ℂ)) x).symm)\n ((ext... | exact ⟨continuous_mul, fun x y => contDiff_mul.contDiffOn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 56
} | [
{
"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 ContinuousAt.comp _ continuousAt_subtype_val | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique | {
"line": 213,
"column": 6
} | {
"line": 213,
"column": 56
} | [
{
"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 ContinuousAt.comp _ continuousAt_subtype_val | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 774,
"column": 4
} | {
"line": 774,
"column": 55
} | [
{
"pp": "case hs\n𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nH' : Type u_5\ninst✝⁷ : ... | exact UniqueMDiffOn.uniqueDiffOn_target_inter hs x₀ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.Riemannian.Basic | {
"line": 475,
"column": 36
} | {
"line": 477,
"column": 53
} | [
{
"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
rw [lintegral_const_mul' _ _ ENNReal.coe_ne_top,
pathELength_eq_lintegral_mfderivWithin_Icc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Riemannian.Basic | {
"line": 450,
"column": 6
} | {
"line": 484,
"column": 86
} | [] | ‖γ' t₁ - γ' 0‖ₑ
_ ≤ ∫⁻ t' in Icc 0 t₁, ‖derivWithin γ' (Icc 0 t₁) t'‖ₑ := by
apply enorm_sub_le_lintegral_derivWithin_Icc_of_contDiffOn_Icc _ ht₁0
rwa [← contMDiffOn_iff_contDiffOn]
_ = ∫⁻ t' in Icc 0 t₁, ‖mfderiv[Icc 0 t₁] γ' t' 1‖ₑ := by
simp_rw [← fderivWithin_derivWithin, mfderivWithin_eq_... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 225,
"column": 86
} | {
"line": 241,
"column": 10
} | [
{
"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... | by
refine ⟨(1 + ‖(trivializationAt F E x).continuousLinearMapAt ℝ x‖) * 2, by positivity, ?_⟩
filter_upwards [eventually_norm_symmL_trivializationAt_self_comp_lt F E x one_lt_two] with y hy
have A : ((trivializationAt F E x).continuousLinearMapAt ℝ x) ∘L
((trivializationAt F E x).symmL ℝ x) = ContinuousLin... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.WhitneyEmbedding | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 79
} | [
{
"pp": "case intro\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : IsManifold I ∞ M\ninst✝¹ : T2Space M\nι :... | letI : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.1 inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 244,
"column": 22
} | {
"line": 246,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁰ : TopologicalSpace M\ninst✝⁹ : ChartedSpace H M\nF : Type u_5\ninst✝⁸ : NormedAdd... | by
simp [hcov.leibniz hσ hφ, hcov'.leibniz hσ hφ]
module | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 127,
"column": 76
} | {
"line": 127,
"column": 84
} | [
{
"pp": "case succ\nn : ℕ\ni : ZMod n\nk : ℕ\nIH : r i ^ k = r (i * ↑k)\n⊢ i * ↑k + i = i * (↑k + 1)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"ZMod.commRing",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PresentedGroup | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 26
} | [
{
"pp": "case mk.mul\nα : Type u_1\nrels : Set (FreeGroup α)\nH : Subgroup (PresentedGroup rels)\nh : ∀ (j : α), of j ∈ H\nx : PresentedGroup rels\nx✝ y✝ : FreeGroup α\nh1 : Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rels))) x✝ ∈ H\nh2 : Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rel... | QuotientGroup.mk_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 56
} | [
{
"pp": "case h₁\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nhω : cs.IsReduced ω\nthis : cs.length (cs.wordProd ω.reverse) ≤ ω.reverse.length\n⊢ cs.length (cs.wordProd ω)⁻¹ ≤ cs.length (cs.wordProd ω)",
"usedConstants": [
"DivInvOneMonoid.toIn... | rwa [wordProd_reverse, length_reverse, ← hω] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 160,
"column": 10
} | {
"line": 160,
"column": 19
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nhω : cs.IsReduced ω\n⊢ cs.lengthParity (cs.wordProd ω) = Multiplicative.ofAdd ↑ω.length",
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"InvOneClass.toOne",
"Equiv.i... | wordProd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 104,
"column": 2
} | {
"line": 106,
"column": 50
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nt : W\nht : cs.IsReflection t\nw : W\n⊢ cs.length (t * w) ≠ cs.length w",
"usedConstants": [
"MonoidHom.instFunLike",
"Equiv.instEquivLike",
"HMul.hMul",
"ZMod.commRing",
"MonoidH... | suffices cs.lengthParity (t * w) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.GroupTheory.CoprodI | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 49
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝² : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝¹ : Monoid N\nG : ι → Type u_4\ninst✝ : (i : ι) → Group (G i)\nm : CoprodI G\n⊢ MulOpposite.unop ((lift fun i ↦ (MonoidHom.op of).comp (MulEquiv.inv' (G i)).toMonoidHom) m) * m = 1",
"usedConstants": [
"Eq.mpr... | induction m using CoprodI.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.GroupTheory.CoprodI | {
"line": 355,
"column": 4
} | {
"line": 355,
"column": 15
} | [
{
"pp": "case neg\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\nm' : M i\nw' : Word M\nh' : w'.fstIdx ≠ some i\nhm : m = 1\nhm' : ¬m' = 1\nh : (cons m' w' ⋯ ⋯).fstIdx ≠ some i\nhe : w = cons m' w' ⋯ ⋯\n⊢ False",
"usedConsta... | exact h rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.CoprodI | {
"line": 459,
"column": 14
} | {
"line": 459,
"column": 49
} | [
{
"pp": "case inr\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\nw : Word M\nm : M i\n⊢ (i ≠ j ∧ ∃ (h : w.toList ≠ []), w.toList.head h = ⟨i, m⟩) → ⟨i, m⟩ ∈ w.toList",
"usedConstants": [
"List.head",
"Iff.mp... | cases w.toList <;> simp +contextual | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.GroupTheory.CoprodI | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 45
} | [
{
"pp": "ι : 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)\nm : CoprodI M\n⊢ ∀ (w : Word M), (m • w).prod = m * w.prod",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"instHSMul",
"Mono... | induction m using CoprodI.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 111,
"column": 15
} | {
"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... | unfold Matrix.IsSymm; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 111,
"column": 15
} | {
"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... | unfold Matrix.IsSymm; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 130,
"column": 15
} | {
"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",
... | unfold Matrix.IsSymm; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 130,
"column": 15
} | {
"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",
... | unfold Matrix.IsSymm; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 152,
"column": 15
} | {
"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",
... | unfold Matrix.IsSymm; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 152,
"column": 15
} | {
"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",
... | unfold Matrix.IsSymm; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 168,
"column": 15
} | {
"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",
... | unfold Matrix.IsSymm; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Matrix | {
"line": 168,
"column": 15
} | {
"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",
... | unfold Matrix.IsSymm; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 822,
"column": 8
} | {
"line": 822,
"column": 23
} | [
{
"pp": "ι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i\ni✝ j✝ i j k✝ l : ι\nw₁ : NeWord H i j\nhne : j ≠ k✝\nw₂ : NeWord... | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.CoprodI | {
"line": 822,
"column": 8
} | {
"line": 822,
"column": 23
} | [
{
"pp": "ι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i\ni✝ j✝ i j k✝ l : ι\nw₁ : NeWord H i j\nhne : j ≠ k✝\nw₂ : NeWord... | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CoprodI | {
"line": 822,
"column": 8
} | {
"line": 822,
"column": 23
} | [
{
"pp": "ι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i\ni✝ j✝ i j k✝ l : ι\nw₁ : NeWord H i j\nhne : j ≠ k✝\nw₂ : NeWord... | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 924,
"column": 16
} | {
"line": 924,
"column": 35
} | [
{
"pp": "case h\nι✝ : Type u_1\nM : ι✝ → Type u_2\ninst✝² : (i : ι✝) → Monoid (M i)\nN : Type u_3\ninst✝¹ : Monoid N\nι : Type u_4\nX : ι → Type u_5\nG : ι → Type u_6\ninst✝ : (i : ι) → Group (G i)\nB : (i : ι) → FreeGroupBasis (X i) (G i)\ni : ι\n⊢ ((FreeGroup.lift fun x ↦ of ((B x.fst) x.snd)).comp (lift fun ... | apply (B i).ext_hom | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.CoprodI | {
"line": 990,
"column": 4
} | {
"line": 990,
"column": 29
} | [
{
"pp": "case hXnonempty\nι : Type u_1\ninst✝² : Nontrivial ι\nG : Type u_1\ninst✝¹ : Group G\na : ι → G\nα : Type u_4\ninst✝ : MulAction G α\nX Y : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhYdisj : Pairwise (Disjoint on Y)\nhXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j... | show ∀ i, (X' i).Nonempty | Lean.Elab.Tactic.evalShow | Lean.Parser.Tactic.show |
Mathlib.GroupTheory.Frattini | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 60
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nhφ : Function.Surjective ⇑φ\n⊢ frattini G ≤ comap φ (frattini H)",
"usedConstants": [
"Eq.mpr",
"iInf",
"Order.radical",
"congrArg",
"PartialOrder.toPreorder",
"setOf",
"Preorder.toL... | simp_rw [frattini, Order.radical, comap_iInf, le_iInf_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.Nilpotent | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 20
} | [
{
"pp": "G : 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 * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ (fun m ↦ H (n - m)) (m + 1)\ng : G\n⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m ↦ H (n - m)) m",
"usedConstants": [
"HMul.h... | dsimp only at hx ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.GroupTheory.FreeGroup.Orbit | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 24
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na b : α × Bool\nh : ∀ (x : FreeGroup α), x.toWord[0]? = some a ↔ x.toWord[0]? = some b\n⊢ a = b",
"usedConstants": [
"congrArg",
"FreeGroup.toWord",
"AddMonoid.toAddZeroClass",
"List.instGetElem?NatLtLength",
"Nat.instAddMonoid",
... | simpa using h (mk [a]) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.Transfer | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 63
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\np : ℕ\nP : Sylow p G\nhP : normalizer ↑↑P ≤ centralizer ↑P\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Finite (Sylow p G)\ninst✝ : (↑P).FiniteIndex\nhf : Function.Bijective ⇑((transferSylow P hP).restrict ↑P)\n⊢ (transferSylow P hP).ker.IsComplement' ↑P",
"usedConstants":... | rw [Function.Bijective, ← range_eq_top, restrict_range] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Nilpotent | {
"line": 516,
"column": 4
} | {
"line": 516,
"column": 33
} | [
{
"pp": "case succ\nG : Type u_1\ninst✝¹ : Group G\nH : Type u_2\ninst✝ : Group H\nf : G →* H\nd : ℕ\nhd : Subgroup.map f (lowerCentralSeries G d) ≤ lowerCentralSeries H d\nx : G\nhx : x ∈ lowerCentralSeries G d.succ\n⊢ ∀ x ∈ {g | ∃ g₁ ∈ lowerCentralSeries G d, ∃ g₂ ∈ ⊤, ⁅g₁, g₂⁆ = g}, f x ∈ lowerCentralSeries ... | rintro a ⟨y, hy, z, ⟨-, rfl⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 266,
"column": 10
} | {
"line": 266,
"column": 62
} | [
{
"pp": "case pos\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' : Labellin... | rw [loopOfHom_eq_id T e h, ← End.one_def, E.map_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 266,
"column": 10
} | {
"line": 266,
"column": 62
} | [
{
"pp": "case pos\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' : Labellin... | rw [loopOfHom_eq_id T e h, ← End.one_def, E.map_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 266,
"column": 10
} | {
"line": 266,
"column": 62
} | [
{
"pp": "case pos\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' : Labellin... | rw [loopOfHom_eq_id T e h, ← End.one_def, E.map_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 178,
"column": 40
} | {
"line": 183,
"column": 44
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\na : X\nha : a ∉ fixedPoints G X\nH : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B\nthis : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\n⊢ IsTrivialBlock B",
"usedConstants": [
"Eq.mpr",
"instHSMul",
... | by
obtain rfl | ⟨b, hb⟩ := B.eq_empty_or_nonempty
· simp [IsTrivialBlock]
· obtain ⟨g, hg⟩ := exists_smul_eq G b a
rw [← IsTrivialBlock.smul_iff g]
exact H ⟨b, hb, hg⟩ (hB.translate g) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 45
} | [
{
"pp": "case hs\nG : Type u_1\nX : Type u_2\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G X\nH : Type u_3\nY : Type u_4\ninst✝⁴ : Group H\ninst✝³ : MulAction H Y\nφ : G → H\nf : X →ₑ[φ] Y\ninst✝² : Finite Y\ninst✝¹ : IsPretransitive H Y\ninst✝ : IsPreprimitive G X\nhf' : Nat.card Y < 2 * (Set.range ⇑f).ncard\nB : Se... | simp only [Set.preimage_eq_univ_iff] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer | {
"line": 251,
"column": 20
} | {
"line": 253,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\ns : Set α\ng k : ↥(stabilizer G s)\nx : ↑s\n⊢ (g * k) • x = g • k • x",
"usedConstants": [
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
"congrArg",
"Membership.mem",
"Set.Elem",
"Subtype",... | by
simp only [← Subtype.coe_inj, SMul.smul_stabilizer_def, Subgroup.coe_mul,
SemigroupAction.mul_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 479,
"column": 6
} | {
"line": 479,
"column": 30
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\nN : Subgroup G\ninst✝ : N.Normal\na : X\ng : G\n⊢ g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a)",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"CompleteLattice.instOmegaCompletePartia... | smul_orbit_eq_orbit_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 509,
"column": 46
} | {
"line": 509,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\nS : Type u_3\nH : Type u_4\ninst✝⁴ : Group H\ninst✝³ : SetLike S H\ninst✝² : SubgroupClass S H\ns : S\ninst✝¹ : MulAction G H\ninst✝ : IsScalarTower G H H\na b : G\nhab : a • ↑s ≠ b • ↑s\nc : H\nhc : c ∈ ↑s\nd : H\nhd : d ∈ ↑s\nhcd : b • d = a • c\n⊢ (b • d) • ↑s = b • c... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 526,
"column": 69
} | {
"line": 526,
"column": 80
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\nS : Type u_3\nH : Type u_4\ninst✝⁴ : Group H\ninst✝³ : SetLike S H\ninst✝² : SubgroupClass S H\ns : S\ninst✝¹ : MulAction G H\ninst✝ : IsScalarTower G Hᵐᵒᵖ H\na b : G\nhab : a • ↑s ≠ b • ↑s\nc : H\nhc : c ∈ ↑s\nd : H\nhd : d ∈ ↑s\nhcd : b • d = a • c\n⊢ (b • op d) • ↑s =... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 52
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : σ ∈ alternatingGroup α\nn : ℕ\nih : ∀ (l : List (Perm α)), (∀ g ∈ l, g.IsSwap) → l.length = 2 * n → l.prod ∈ closure {σ | σ.IsThreeCycle}\na : Perm α\nl : List (Perm α)\nhl : ∀ g ∈ a :: l, g.IsSwap\nhn : l.length = 2 *... | obtain ⟨b, l, rfl⟩ := l.exists_of_length_succ hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 708,
"column": 8
} | {
"line": 708,
"column": 17
} | [
{
"pp": "case inr.«0»\nG : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\ninst✝ : IsPretransitive G X\nB : Set X\nhX : Finite X\nhB : IsBlock G B\nhB_ne : B.Nonempty\nhB' : 0 < 2\nkey : B.ncard * 0 = Nat.card X\n⊢ B = Set.univ",
"usedConstants": [
"Nat.instMulZeroClass",
"HMul... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 330,
"column": 6
} | {
"line": 331,
"column": 96
} | [
{
"pp": "h3 : (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)⁻¹ * (finRotate 5)⁻¹).IsThreeCycle\nh : ⟨finRotate 5, ⋯⟩ ∈ normalClosure {⟨finRotate 5, ⋯⟩}\n⊢ ⟨Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)⁻¹ * (finRotate 5)⁻¹, ⋯⟩ ∈ normalClosure {⟨finRotate 5, ⋯⟩}",
"usedConstants": [
"instNeZ... | exact (mul_mem (Subgroup.normalClosure_normal.conj_mem _ h
⟨Fin.cycleRange 2, Fin.isThreeCycle_cycleRange_two.mem_alternatingGroup⟩) (inv_mem h) :) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 53
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nG : Subgroup (Perm α)\nhG : G.index = 2\na✝ : Nontrivial α\na b : α\nhab : a ≠ b\ng✝ g : Perm α\nx y : α\nhxy : x ≠ y\nih : g ∈ G ↔ g ∈ alternatingGroup α\nhabG : swap x y ∈ G\n⊢ ⋯.choose * swap x y * ⋯.choose⁻¹ ∈ G",
"usedConstants":... | exact (normal_of_index_eq_two hG).conj_mem _ habG _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 18
} | [
{
"pp": "case h.h\nG : Type u_1\nα : Type u_2\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\nH : Type u_3\nβ : Type u_4\ninst✝² : Group H\ninst✝¹ : MulAction H β\nσ : G → H\nf : α →ₑ[σ] β\nn : Type u_5\nhf : Function.Surjective ⇑f\ninst✝ : IsPretransitive G (n ↪ α)\nx y : n ↪ β\naux : (n ↪ β) → n ↪ α := fun x ↦ x.t... | smul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 417,
"column": 48
} | {
"line": 417,
"column": 56
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G α\nm n : ℕ\nHn : IsMultiplyPretransitive G α n\ns : Set α\ninst✝ : Finite ↑s\nhmn : s.ncard + m = n\nx y : Fin m ↪ ↥(ofFixingSubgroup G s)\nthis : IsMultiplyPretransitive G α (s.ncard + m)\nHs : Nonempty (Fin s.ncard ≃ ↑s)\nx' : Fin (s.... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 244,
"column": 6
} | {
"line": 244,
"column": 44
} | [
{
"pp": "case h.left\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\ng : M\nhg : g • t = s\nk : M\nhk : k ∈ fixingSubgroup M s\n⊢ (MulAut.conj g⁻¹) k ∈ fixingSubgroup M t",
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",
"Group.toDivisionMonoid",
"Div... | apply Set.conj_mem_fixingSubgroup _ hk | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 371,
"column": 6
} | {
"line": 371,
"column": 29
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\na : α\nha : a ∈ ofFixingSubgroup M s\nha' : ⟨a, ha⟩ ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)\nthis : a ∈ ofFixingSubgroup M (s ∪ t)\n⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = ⟨⟨a, ha⟩, ha'⟩",
... | exact ⟨⟨a, this⟩, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 371,
"column": 6
} | {
"line": 371,
"column": 29
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\na : α\nha : a ∈ ofFixingSubgroup M s\nha' : ⟨a, ha⟩ ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)\nthis : a ∈ ofFixingSubgroup M (s ∪ t)\n⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = ⟨⟨a, ha⟩, ha'⟩",
... | exact ⟨⟨a, this⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 371,
"column": 6
} | {
"line": 371,
"column": 29
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\na : α\nha : a ∈ ofFixingSubgroup M s\nha' : ⟨a, ha⟩ ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)\nthis : a ∈ ofFixingSubgroup M (s ∪ t)\n⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = ⟨⟨a, ha⟩, ha'⟩",
... | exact ⟨⟨a, this⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 215,
"column": 6
} | {
"line": 218,
"column": 29
} | [
{
"pp": "n : ℕ\nhrec :\n ∀ m < n,\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 = m + 1 →\n m + 2 < Nat.card α →\n (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → Is... | refine ⟨this, fun hs_prim ↦ ?_⟩
apply (hrec _ hmn hG htm htm').2
exact IsPreprimitive.isPreprimitive_ofFixingSubgroup_inter
hs_prim hsgs_ne_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 215,
"column": 6
} | {
"line": 218,
"column": 29
} | [
{
"pp": "n : ℕ\nhrec :\n ∀ m < n,\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 = m + 1 →\n m + 2 < Nat.card α →\n (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → Is... | refine ⟨this, fun hs_prim ↦ ?_⟩
apply (hrec _ hmn hG htm htm').2
exact IsPreprimitive.isPreprimitive_ofFixingSubgroup_inter
hs_prim hsgs_ne_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 34
} | [
{
"pp": "case inl\nM : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\nn : ℕ\nhα : ↑n.succ ≤ ENat.card α\ninst✝ : IsMultiplyPretransitive M α n.succ\nhn : n = 0\n⊢ IsMultiplyPreprimitive M α 0",
"usedConstants": [
"MulAction.is_zero_preprimitive"
]
}
] | exact is_zero_preprimitive M α | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 38
} | [
{
"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✝¹ : Finite α\nthis✝ : Fintype α\nφ : K →* Perm α := toPermHom K α\nf : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }\nhf : Function.Bijective ⇑f\nthis : Function... | apply Perm.isMultiplyPretransitive | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 715,
"column": 2
} | {
"line": 715,
"column": 44
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nB : Set α\nhB : IsBlock (↥(alternatingGroup α)) B\nh2 : Nat.card α ≤ 2\n⊢ IsTrivialBlock B",
"usedConstants": [
"Finite.of_fintype",
"MulAction.isTrivialBlock_of_card_le_two"
]
},
{
"pp": "case inr\nα : Type ... | · exact isTrivialBlock_of_card_le_two h2 B | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.GroupExtension.Defs | {
"line": 202,
"column": 49
} | {
"line": 202,
"column": 70
} | [
{
"pp": "N : Type u_1\nE : Type u_2\nG : Type u_3\ninst✝³ : Group N\ninst✝² : Group E\ninst✝¹ : Group G\nS : GroupExtension N E G\nE' : Type u_4\ninst✝ : Group E'\nS' : GroupExtension N E' G\nequiv : S.Equiv S'\n⊢ ⇑S.rightHom = ⇑S'.rightHom ∘ ⇑equiv.toMulEquiv",
"usedConstants": [
"Eq.mpr",
"Mul... | ← equiv.rightHom_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.IsPerfect | {
"line": 68,
"column": 69
} | {
"line": 68,
"column": 83
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\n⊢ ⁅⊤, ⊤⁆ = ⊤ ↔ _root_.commutator G = ⊤",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Bracket.bracket",
"id",
"Subgroup",
"commutator",
"Iff",
"Subgroup.instTop",
"commutator_def",
"Top.top",
"Subgroup.com... | commutator_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.IsPerfect | {
"line": 66,
"column": 62
} | {
"line": 68,
"column": 84
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\n⊢ IsPerfect ↥⊤ ↔ IsPerfect G",
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"Subgroup.map_subtype_inj",
"Subgroup.map",
"congrArg",
"Subgroup.subtype",
"Iff.rfl",
"Group.isPerfect_def",
"Bracket.bracket",
... | by
rw [isPerfect_def, isPerfect_def, ← map_subtype_inj,
map_subtype_commutator, ← MonoidHom.range_eq_map, subtype_range, commutator_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.DomMulAct | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 26
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nf : α → ι\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq ι\nφ : α → ↥(Finset.image f Finset.univ) := Set.codRestrict f ↑(Finset.image f Finset.univ) ⋯\ng : Perm α\n⊢ f ∘ ⇑g = f ↔ φ ∘ ⇑g = φ",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike... | simp only [funext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nc : ↥g.cycleFactorsFinset\na✝¹ : c ∈ ↑(↑τ).support\nd : ↥g.cycleFactorsFinset\na✝ : d ∈ ↑(↑τ).support\nh : c ≠ d\n⊢ Function.onFun _root_.Disjoint (fun c ↦ (↑c).support) c d",
"usedConstants... | apply Equiv.Perm.Disjoint.disjoint_support | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 145,
"column": 53
} | {
"line": 152,
"column": 35
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : DecidableEq α\ninst✝ : Finite α\nS : Set G\nhS1 : ∀ σ ∈ S, (toPermHom G α) σ = 1 ∨ ((toPermHom G α) σ).IsSwap\nhS2 : closure S = ⊤\nh : IsPretransitive G α\n⊢ Function.Surjective ⇑(toPermHom G α)",
"usedConstants": [
... | by
have h : closure ((toPermHom G α '' S) \ {1}) = (toPermHom G α).range := by
rw [closure_diff_one, ← MonoidHom.map_closure, hS2, ← MonoidHom.range_eq_map]
have := IsPretransitive.of_compHom (α := α) (toPermHom G α).rangeRestrict
rw [← h] at this
rw [← MonoidHom.range_eq_top, ← h, closure_of_isSwap_of_isPr... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.RegularWreathProduct | {
"line": 255,
"column": 8
} | {
"line": 255,
"column": 77
} | [
{
"pp": "D : Type u_1\nQ : Type u_2\ninst✝⁵ : Group D\ninst✝⁴ : Group Q\nG✝ : Type u\nn✝ : ℕ\ninst✝³ : Group G✝\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nα : Type u_3\ninst✝² : Finite α\nhα : Nat.card α = p ^ n\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : Finite G\nhG : Nat.card G = p\nP : Sylow p (Equiv.Perm α)\ne1 ... | ← Nat.multiplicity_eq_factorization hp.out (p ^ n).factorial_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 477,
"column": 34
} | {
"line": 477,
"column": 46
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\ng : ↥A\nw : NormalWord d\nthis : Cancels 1 (↑g • w) ↔ Cancels 1 w\nhcan : ¬Cancels 1 w\n⊢ (let g' := unitsSMulGroup φ d 1 (↑g • w).head;\n cons (↑g'.1) 1 ((↑g'.2 * (↑g • w).head⁻¹) • ↑g • w) ⋯ ⋯) =\n ... | dif_neg hcan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 530,
"column": 6
} | {
"line": 538,
"column": 45
} | [
{
"pp": "case pos.cons\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\ng✝ : G\nu✝ : ℤˣ\nw✝ : NormalWord d\nh1✝ : w✝.head ∈ d.set u✝\nh2✝ : ∀ u' ∈ Option.map Prod.fst w✝.toList.head?, w✝.head ∈ toSubgroup A B u✝ → u✝ = u'\nhcan : Cancels u (cons g✝ u✝ w✝ h1✝ h2✝... | cases hcan.2
simp only [unitsSMulWithCancel, id_eq, consRecOn_cons, prod_group_smul, prod_cons, zpow_neg]
rcases Int.units_eq_one_or u with (rfl | rfl)
· simp [equiv_eq_conj, mul_assoc]
· -- Before https://github.com/leanprover/lean4/pull/2644, this proof was just
-- simp [equiv_symm_eq_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.HNNExtension | {
"line": 530,
"column": 6
} | {
"line": 538,
"column": 45
} | [
{
"pp": "case pos.cons\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\ng✝ : G\nu✝ : ℤˣ\nw✝ : NormalWord d\nh1✝ : w✝.head ∈ d.set u✝\nh2✝ : ∀ u' ∈ Option.map Prod.fst w✝.toList.head?, w✝.head ∈ toSubgroup A B u✝ → u✝ = u'\nhcan : Cancels u (cons g✝ u✝ w✝ h1✝ h2✝... | cases hcan.2
simp only [unitsSMulWithCancel, id_eq, consRecOn_cons, prod_group_smul, prod_cons, zpow_neg]
rcases Int.units_eq_one_or u with (rfl | rfl)
· simp [equiv_eq_conj, mul_assoc]
· -- Before https://github.com/leanprover/lean4/pull/2644, this proof was just
-- simp [equiv_symm_eq_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 52,
"column": 2
} | {
"line": 57,
"column": 19
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh : Subgroup.centralizer {g} ≤ alternatingGroup α\nc : Perm α\nhc : c ∈ g.cycleFactorsFinset\n⊢ Odd ((card ∘ support) c)",
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"MonoidHom.instFunLike",
"NonUni... | suffices sign c = 1 by
rw [IsCycle.sign _, neg_eq_iff_eq_neg, ← Int.units_ne_iff_eq_neg] at this
· rw [← Nat.not_even_iff_odd, comp_apply]
exact fun h ↦ this h.neg_one_pow
· rw [mem_cycleFactorsFinset_iff] at hc
exact hc.left | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.GroupTheory.HNNExtension | {
"line": 634,
"column": 2
} | {
"line": 634,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : ReducedWord G A B\ng : G\nl : List (ℤˣ × G)\nchain : List.IsChain (fun a b ↦ a.2 ∈ toSubgroup A B a.1 → a.1 = b.1) l\nhw1 : { head := g, toList := l, chain := chain }.head = 1\n⊢ ∃ w',\n ReducedWord.prod φ ... | dsimp at hw1 | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 47
} | [
{
"pp": "G : 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'\nh3 : ∀ (H : S... | rwa [map_subtype_inj, map_subtype_inj] at key | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\n⊢ ∀ (g : ↥(kleinFour α)), orderOf g ∣ 2",
"usedConstants": [
"Membership.mem",
"Subtype",
"Subgroup",
"Equiv.Perm.permGroup",
"alternatingGroup.kleinFour",
"Equiv.Perm",
"alterna... | rintro ⟨⟨g, hg⟩, hg'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 228,
"column": 13
} | {
"line": 228,
"column": 22
} | [
{
"pp": "case inl\n⊢ orderOf (a 1) = 2 * 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"DivInvMonoid.toMonoid",
"instMulNat",
"instOfNatNat",
"ZM... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 86
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nhf : Function.Injective ⇑f\np : ℕ\nhp : Nat.Prime p\nP : Sylow p G\nQ : Sylow p G'\nhQ : Subgroup.comap f ↑Q = ↑P\nhG' : IsCyclic ↥↑Q\nh : Subgroup.map f ↑P ≤ ↑Q\nthis : IsCyclic ↥(Subgroup.map f ↑P)\n⊢ IsCyclic ↥↑P",
"us... | exact isCyclic_of_surjective _ (Subgroup.equivMapOfInjective P f hf).symm.surjective | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 145,
"column": 6
} | {
"line": 145,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsZGroup G\n⊢ IsCyclic ↥(commutator G)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Bracket.bracket",
"Membership.mem",
"id",
"Subtype",
"Subgroup",
"commutator",
"Subgroup.zpow",
"... | commutator_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 148,
"column": 2
} | {
"line": 149,
"column": 18
} | [
{
"pp": "case ind.inl\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsZGroup G\nhH : ∀ y < ⊥, IsCyclic ↥⁅y, y⁆\n⊢ IsCyclic ↥⁅⊥, ⊥⁆",
"usedConstants": [
"Eq.mpr",
"Bot.isCyclic",
"congrArg",
"Bracket.bracket",
"Membership.mem",
"inferInstance",
"id",
... | · rw [Subgroup.commutator_bot_left]
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 244,
"column": 13
} | {
"line": 244,
"column": 27
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝⁴ : Group G\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\ninst✝² : Finite G\nP : Sylow p G\ninst✝¹ : IsCyclic ↥↑P\ninst✝ : (↑P).Normal\nK : Subgroup G\nhK : K.IsComplement' ↑P\nh : ⁅K, ↑P⁆ = ↑P\n⊢ ⁅K, ↑P⁆ ≤ commutator G",
"usedConstants": [
"Sylow.toSubgroup",
"... | commutator_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 58
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Finite G\nP : Sylow p G\ninst✝ : IsCyclic ↥↑P\nQ : Sylow p ↥(Subgroup.normalizer ↑↑P) := P.subtype ⋯\nthis✝ : (↑Q).Normal\nthis : IsCyclic ↥↑Q\nh : ↑P ≤ ⁅Subgroup.normalizer ↑↑P, Subgroup.normalizer ↑↑P⁆\n⊢ ↑P ≤... | exact h.trans (Subgroup.commutator_mono le_top le_top) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.PushoutI | {
"line": 678,
"column": 8
} | {
"line": 680,
"column": 34
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : Group H\nφ : (i : ι) → H →* G i\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp only [w, Word.prod, List.map_cons, List.prod_cons, List.prod_nil,
List.map_nil, map_mul, ofCoprodI_of, hg₁, hg₂, map_inv, mul_one,
mul_inv_cancel, one_mem] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.PushoutI | {
"line": 678,
"column": 8
} | {
"line": 680,
"column": 34
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : Group H\nφ : (i : ι) → H →* G i\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp only [w, Word.prod, List.map_cons, List.prod_cons, List.prod_nil,
List.map_nil, map_mul, ofCoprodI_of, hg₁, hg₂, map_inv, mul_one,
mul_inv_cancel, one_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.PushoutI | {
"line": 678,
"column": 8
} | {
"line": 680,
"column": 34
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : Group H\nφ : (i : ι) → H →* G i\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp only [w, Word.prod, List.map_cons, List.prod_cons, List.prod_nil,
List.map_nil, map_mul, ofCoprodI_of, hg₁, hg₂, map_inv, mul_one,
mul_inv_cancel, one_mem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Basic | {
"line": 60,
"column": 17
} | {
"line": 63,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nf : α → β\nhf : Measurable f\n⊢ Measurable fun a ↦ Measure.dirac (f a)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Set.in... | by
refine Measure.measurable_of_measurable_coe _ fun s hs => ?_
simp_rw [Measure.dirac_apply' _ hs]
exact measurable_one.indicator (hf hs) | [anonymous] | Lean.Parser.Term.byTactic |
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