module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.IntegralCurve.Basic | {
"line": 92,
"column": 4
} | {
"line": 95,
"column": 57
} | {
"line": 96,
"column": 2
} | [
{
"pp": "case mp\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nH : Type u_2\ninst✝² : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\nt₀ : ℝ\n⊢ IsMIntegralCurveAt γ v t₀ → ∃ ... | [] | intro h
rw [IsMIntegralCurveAt, Filter.eventually_iff_exists_mem] at h
obtain ⟨s, hs, h⟩ := h
exact ⟨s, hs, fun t ht ↦ (h t ht).hasMFDerivWithinAt⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IntegralCurve.Basic | {
"line": 92,
"column": 4
} | {
"line": 95,
"column": 57
} | {
"line": 96,
"column": 2
} | [
{
"pp": "case mp\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nH : Type u_2\ninst✝² : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\nt₀ : ℝ\n⊢ IsMIntegralCurveAt γ v t₀ → ∃ ... | [] | intro h
rw [IsMIntegralCurveAt, Filter.eventually_iff_exists_mem] at h
obtain ⟨s, hs, h⟩ := h
exact ⟨s, hs, fun t ht ↦ (h t ht).hasMFDerivWithinAt⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 478,
"column": 62
} | {
"line": 478,
"column": 71
} | {
"line": 478,
"column": 71
} | [
{
"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... | [] | exact D0' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 478,
"column": 62
} | {
"line": 478,
"column": 71
} | {
"line": 478,
"column": 71
} | [
{
"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... | [] | exact D0' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 478,
"column": 62
} | {
"line": 478,
"column": 71
} | {
"line": 478,
"column": 71
} | [
{
"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... | [] | exact D0' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 517,
"column": 61
} | {
"line": 517,
"column": 70
} | {
"line": 517,
"column": 70
} | [
{
"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... | [] | exact D0' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 517,
"column": 61
} | {
"line": 517,
"column": 70
} | {
"line": 517,
"column": 70
} | [
{
"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... | [] | exact D0' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 517,
"column": 61
} | {
"line": 517,
"column": 70
} | {
"line": 517,
"column": 70
} | [
{
"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... | [] | exact D0' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.UnitInterval | {
"line": 93,
"column": 55
} | {
"line": 94,
"column": 70
} | {
"line": 96,
"column": 0
} | [
{
"pp": "x : ↑I\n⊢ volume (Ioi x) = ENNReal.ofReal (1 - ↑x)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Real",
"Set.Ioi",
"MeasureTheory.Measure",
"unitInterval.volume_apply",
"Real.instZero",
"ENNReal.ofReal",
"congrArg",
... | [] | by
simp only [volume_apply, image_subtype_val_Icc_Ioi, Real.volume_Ioc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IntegralCurve.UniformTime | {
"line": 145,
"column": 4
} | {
"line": 155,
"column": 46
} | {
"line": 157,
"column": 0
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : IsManifold I 1 M\ninst✝¹ : T2Space M\nγ γ' : ℝ → M\nv : (x : M) → Tangen... | [] | have ht' := ht
rw [mem_union, or_iff_not_imp_left] at ht
rw [piecewise, if_neg hmem]
apply hγ' t (ht hmem) |>.hasMFDerivAt (Ioo_mem_nhds (ht hmem).1 (ht hmem).2)
|>.hasMFDerivWithinAt (s := Ioo a b ∪ Ioo a' b')
|>.congr_of_eventuallyEq _ (by rw [piecewise, if_neg hmem])
rw [Filter.eventually... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IntegralCurve.UniformTime | {
"line": 145,
"column": 4
} | {
"line": 155,
"column": 46
} | {
"line": 157,
"column": 0
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : IsManifold I 1 M\ninst✝¹ : T2Space M\nγ γ' : ℝ → M\nv : (x : M) → Tangen... | [] | have ht' := ht
rw [mem_union, or_iff_not_imp_left] at ht
rw [piecewise, if_neg hmem]
apply hγ' t (ht hmem) |>.hasMFDerivAt (Ioo_mem_nhds (ht hmem).1 (ht hmem).2)
|>.hasMFDerivWithinAt (s := Ioo a b ∪ Ioo a' b')
|>.congr_of_eventuallyEq _ (by rw [piecewise, if_neg hmem])
rw [Filter.eventually... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 32
} | {
"line": 142,
"column": 2
} | [
{
"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 : (x : M) → ENorm (TangentSpace I x)\na b : ℝ\nγ : ℝ → M\ninst✝ : ∀ (x : M), ENormSM... | [
"case inl\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)\na : ℝ\nγ : ℝ → M\ninst✝ : ∀ (x : M), ENormSMulCl... | rcases h.eq_or_lt with rfl | h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 32
} | {
"line": 174,
"column": 2
} | [
{
"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 : (x : M) → ENorm (TangentSpace I x)\na b : ℝ\nγ : ℝ → M\ninst✝ : ∀ (x : M), ENormSM... | [
"case inl\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)\na : ℝ\nγ : ℝ → M\ninst✝ : ∀ (x : M), ENormSMulCl... | rcases h.eq_or_lt with rfl | h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 31
} | {
"line": 297,
"column": 4
} | [
{
"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 : (x : M) → ENorm (TangentSpace I x)\ninst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSp... | [
"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 : (x : M) → ENorm (TangentSpace I x)\ninst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)\nx ... | rw [← A a haa', ← B b hb'b] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.Riemannian.Basic | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 39
} | {
"line": 249,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : RiemannianBundle fun x ↦ TangentSpace I x\ninst✝¹ : IsManifold I 1 M\ninst✝ : IsCo... | [
"E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : RiemannianBundle fun x ↦ TangentSpace I x\ninst✝¹ : IsManifold I 1 M\ninst✝ : IsContinuousRiem... | filter_upwards [hC, hx] with y hy h'y | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 723,
"column": 4
} | {
"line": 725,
"column": 53
} | {
"line": 726,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nH' : Type u_5\ninst✝⁷ : Topologic... | [
"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nH' : Type u_5\ninst✝⁷ : TopologicalSpace H'\n... | · apply nhdsWithin_le_nhds
filter_upwards [mfderivWithin_eventually_congr_set (I := I) (I' := I') (f := f) s'_eq]
with y hy using by simp [mpullbackWithin, hy] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 726,
"column": 4
} | {
"line": 728,
"column": 53
} | {
"line": 730,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nH' : Type u_5\ninst✝⁷ : Topologic... | [] | · apply nhdsWithin_le_nhds
filter_upwards [mfderivWithin_eventually_congr_set (I := I) (I' := I') (f := f) s'_eq]
with y hy using by simp [mpullbackWithin, hy] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.Riemannian.Basic | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 39
} | {
"line": 270,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : RiemannianBundle fun x ↦ TangentSpace I x\ninst✝¹ : IsManifold I 1 M\ninst✝ : IsCo... | [
"E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : RiemannianBundle fun x ↦ TangentSpace I x\ninst✝¹ : IsManifold I 1 M\ninst✝ : IsContinuousRiem... | filter_upwards [hC, hx] with y hy h'y | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Geometry.Manifold.Sheaf.Smooth | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 36
} | {
"line": 346,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝²⁶ : NontriviallyNormedField 𝕜\nEM : Type u_2\ninst✝²⁵ : NormedAddCommGroup EM\ninst✝²⁴ : NormedSpace 𝕜 EM\nHM : Type u_3\ninst✝²³ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\nE : Type u_4\ninst✝²² : NormedAddCommGroup E\ninst✝²¹ : NormedSpace 𝕜 E\nH : Ty... | [
"case refine_2\n𝕜 : Type u_1\ninst✝²⁶ : NontriviallyNormedField 𝕜\nEM : Type u_2\ninst✝²⁵ : NormedAddCommGroup EM\ninst✝²⁴ : NormedSpace 𝕜 EM\nHM : Type u_3\ninst✝²³ : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\nE : Type u_4\ninst✝²² : NormedAddCommGroup E\ninst✝²¹ : NormedSpace 𝕜 E\nH : Type u_5\ninst... | · apply smoothSheafCommRing.evalAt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 847,
"column": 31
} | {
"line": 847,
"column": 67
} | {
"line": 848,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁷ : TopologicalSpace H\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : IsManifold I (minSmoothness ... | [] | grw [hm', ← hmn, ← le_minSmoothness] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.GroupTheory.Commutator.Finite | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 58
} | {
"line": 103,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\n⊢ Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) ≠ 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"commutatorSet",
"Eq.mpr",
"congrArg",
"closureCommutatorRepresentatives"... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\n⊢ Nat.card ↑(commutatorSet G) ≠ 0"
] | rw [card_commutatorSet_closureCommutatorRepresentatives] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.CommutingProbability | {
"line": 45,
"column": 58
} | {
"line": 45,
"column": 71
} | {
"line": 45,
"column": 72
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Mul M\nM' : Type u_2\ninst✝ : Mul M'\n⊢ ↑(Nat.card { p // Commute p.1 p.2 }) / ↑(Nat.card M * Nat.card M') ^ 2 =\n ↑(Nat.card { p // Commute p.1 p.2 }) * ↑(Nat.card { p // Commute p.1 p.2 }) /\n (↑(Nat.card M) ^ 2 * ↑(Nat.card M') ^ 2)",
"ppTerm": "?m.12",
"assign... | [
"M : Type u_1\ninst✝¹ : Mul M\nM' : Type u_2\ninst✝ : Mul M'\n⊢ ↑(Nat.card { p // Commute p.1 p.2 }) / (↑(Nat.card M) * ↑(Nat.card M')) ^ 2 =\n ↑(Nat.card { p // Commute p.1 p.2 }) * ↑(Nat.card { p // Commute p.1 p.2 }) /\n (↑(Nat.card M) ^ 2 * ↑(Nat.card M') ^ 2)"
] | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 94,
"column": 56
} | {
"line": 94,
"column": 69
} | {
"line": 94,
"column": 70
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\n⊢ ↑(Nat.card (ConjClasses G) * Nat.card G) / ↑(Nat.card G) ^ 2 = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"HMu... | [
"G : Type u_2\ninst✝ : Group G\n⊢ ↑(Nat.card (ConjClasses G)) * ↑(Nat.card G) / ↑(Nat.card G) ^ 2 = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Congruence.Star | {
"line": 30,
"column": 4
} | {
"line": 30,
"column": 39
} | {
"line": 32,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Mul M\ninst✝ : StarMul M\nr : M → M → Prop\nhr : ∀ (a b : M), r a b → r (Star.star a) (Star.star b)\na b w✝ x✝ y✝ z✝ : M\nh1 : Rel r w✝ x✝\nh2 : Rel r y✝ z✝\n⊢ Rel r (Star.star y✝ * Star.star w✝) (Star.star z✝ * Star.star x✝)",
"ppTerm": "?m.312",
"assigned": true,
"u... | [] | exact (h2.star hr).mul (h1.star hr) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 610,
"column": 48
} | {
"line": 610,
"column": 67
} | {
"line": 610,
"column": 67
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\n⊢ inl.range ⊔ inr.range = inl.range ⊔ Subgroup.closure ↑inr.range",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Subgroup.closure_eq",
"Eq.mpr",
"MonoidHom.range",
"Lattice.toSemilatticeSup",
... | [
"G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\n⊢ inl.range ⊔ inr.range = inl.range ⊔ inr.range"
] | Subgroup.closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 485,
"column": 6
} | {
"line": 485,
"column": 73
} | {
"line": 486,
"column": 6
} | [
{
"pp": "case neg\nB : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm : ℕ\nih :\n cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)\nhm : ¬Even m\nh₁ : Even (m + 1)\n⊢ cs.simple (if Even m then... | [
"case neg\nB : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm : ℕ\nih :\n cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)\nhm : ¬Even m\nh₁ : Even (m + 1)\nh₂ : (m + 1) / 2 = m / 2 + 1\n⊢ cs.simp... | have h₂ : (m + 1) / 2 = m / 2 + 1 := Nat.succ_div_of_dvd h₁.two_dvd | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 498,
"column": 19
} | {
"line": 498,
"column": 24
} | {
"line": 498,
"column": 24
} | [
{
"pp": "B : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm : ℕ\nk : ℤ\n⊢ Even (2 * k)",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Int",
"Int.instMul",
"instHAdd",
"instOfNat",
"HAdd.hA... | [
"case h\nB : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm : ℕ\nk : ℤ\n⊢ 2 * k = k + k"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 41
} | {
"line": 170,
"column": 2
} | [
{
"pp": "case h₁\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\n⊢ cs.length (cs.simple i) ≤ 1",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid",
"Eq.mp",
... | [] | simpa using cs.length_wordProd_le [i] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 41
} | {
"line": 170,
"column": 2
} | [
{
"pp": "case h₁\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\n⊢ cs.length (cs.simple i) ≤ 1",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid",
"Eq.mp",
... | [] | simpa using cs.length_wordProd_le [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 41
} | {
"line": 170,
"column": 2
} | [
{
"pp": "case h₁\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\n⊢ cs.length (cs.simple i) ≤ 1",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid",
"Eq.mp",
... | [] | simpa using cs.length_wordProd_le [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 599,
"column": 6
} | {
"line": 599,
"column": 31
} | {
"line": 600,
"column": 4
} | [
{
"pp": "case empty\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\n⊢ empty.prod • empty = empty",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.to... | [] | rw [prod_empty, one_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 743,
"column": 4
} | {
"line": 743,
"column": 25
} | {
"line": 745,
"column": 0
} | [
{
"pp": "case append\nι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\ni j i✝ j✝ k✝ l✝ : ι\n_w₁✝ : NeWord M i✝ j✝\n_hne✝ : j✝ ≠ k✝\n_w₂✝ : NeWord M k✝ l✝\n_w₁_ih✝ : ∀ (x : M i✝) (hnotone : x ≠ 1), (replaceHead x hnotone _w₁✝).head = x\n_w₂_ih✝ : ∀ (x : M k✝) (hnotone : x ≠ 1), (replaceHead x hnot... | [] | simp [*, replaceHead] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.CoprodI | {
"line": 743,
"column": 4
} | {
"line": 743,
"column": 25
} | {
"line": 745,
"column": 0
} | [
{
"pp": "case append\nι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\ni j i✝ j✝ k✝ l✝ : ι\n_w₁✝ : NeWord M i✝ j✝\n_hne✝ : j✝ ≠ k✝\n_w₂✝ : NeWord M k✝ l✝\n_w₁_ih✝ : ∀ (x : M i✝) (hnotone : x ≠ 1), (replaceHead x hnotone _w₁✝).head = x\n_w₂_ih✝ : ∀ (x : M k✝) (hnotone : x ≠ 1), (replaceHead x hnot... | [] | simp [*, replaceHead] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CoprodI | {
"line": 743,
"column": 4
} | {
"line": 743,
"column": 25
} | {
"line": 745,
"column": 0
} | [
{
"pp": "case append\nι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\ni j i✝ j✝ k✝ l✝ : ι\n_w₁✝ : NeWord M i✝ j✝\n_hne✝ : j✝ ≠ k✝\n_w₂✝ : NeWord M k✝ l✝\n_w₁_ih✝ : ∀ (x : M i✝) (hnotone : x ≠ 1), (replaceHead x hnotone _w₁✝).head = x\n_w₂_ih✝ : ∀ (x : M k✝) (hnotone : x ≠ 1), (replaceHead x hnot... | [] | simp [*, replaceHead] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 407,
"column": 8
} | {
"line": 409,
"column": 59
} | {
"line": 410,
"column": 4
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\n⊢ List.map (fun x ↦ x⁻¹) (cs.rightInvSeq ω.reverse) = List.map id (cs.rightInvSeq ω.reverse)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",... | [] | apply List.map_congr_left
intro t ht
exact (cs.isReflection_of_mem_rightInvSeq _ ht).inv | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 407,
"column": 8
} | {
"line": 409,
"column": 59
} | {
"line": 410,
"column": 4
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\n⊢ List.map (fun x ↦ x⁻¹) (cs.rightInvSeq ω.reverse) = List.map id (cs.rightInvSeq ω.reverse)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",... | [] | apply List.map_congr_left
intro t ht
exact (cs.isReflection_of_mem_rightInvSeq _ ht).inv | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 979,
"column": 12
} | {
"line": 979,
"column": 30
} | {
"line": 979,
"column": 30
} | [
{
"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)... | [
"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), a i • (Y i... | MonoidHom.coe_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 51
} | {
"line": 162,
"column": 2
} | [
{
"pp": "case h\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nx h k : G\nh3 : h ∈ H\nh4 : k ∈ K\nh5 : Quotient.out (mk H K x) = h * x * k\n⊢ ∃ x_1 ∈ H, ∃ y ∈ K, x = x_1 * Quotient.out (mk H K x) * y",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.... | [
"case h\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nx h k : G\nh3 : h ∈ H\nh4 : k ∈ K\nh5 : Quotient.out (mk H K x) = h * x * k\n⊢ x = h⁻¹ * Quotient.out (mk H K x) * k⁻¹"
] | refine ⟨h⁻¹, H.inv_mem h3, k⁻¹, K.inv_mem h4, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.CoprodI | {
"line": 1049,
"column": 2
} | {
"line": 1049,
"column": 26
} | {
"line": 1050,
"column": 2
} | [
{
"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)... | [
"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), a i • (Y i... | show _ ∨ ∃ i, 3 ≤ #(H i) | Lean.Elab.Tactic.evalShow | Lean.Parser.Tactic.show |
Mathlib.Order.Radical | {
"line": 49,
"column": 11
} | {
"line": 49,
"column": 21
} | {
"line": 49,
"column": 21
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : IsCoatomic α\na : α\nh : a ⊔ radical α = ⊤\nm : α\nc : IsCoatom m\nle : a ≤ m\nq : ⊤ ≤ m\n⊢ a = ⊤",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"case inr\nα : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : IsCoatomic α\na : α\nh : a ⊔ radical α = ⊤\nm : α\nc : IsCoatom m\nle : a ≤ m\nq : m = ⊤\n⊢ a = ⊤"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Focal | {
"line": 162,
"column": 2
} | {
"line": 163,
"column": 22
} | {
"line": 164,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.FiniteIndex\nx : ↥H\n⊢ H.transferFocal ↑x = ↑x ^ H.index",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Quotient.finite",
"Fintype.ofFinite",
"Subgroup.finite_quotient_of_finiteIndex",
"Membership.... | [
"G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.FiniteIndex\nx : ↥H\nthis : Fintype (Quotient (MulAction.orbitRel (↥(zpowers ↑x)) (G ⧸ H)))\n⊢ H.transferFocal ↑x = ↑x ^ H.index"
] | have : Fintype (Quotient (MulAction.orbitRel (zpowers (x : G)) (G ⧸ H))) :=
Fintype.ofFinite _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 75,
"column": 4
} | {
"line": 80,
"column": 23
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\nn✝ n : ℕ\nhead : α × Bool\ntail : List (α × Bool)\nh : IsCyclicallyReduced (head :: tail)\n⊢ IsCyclicallyReduced (replicate (n + 1) (head :: tail)).flatten",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"List.head?",
"Eq.mpr",
"F... | [] | rw [isCyclicallyReduced_iff, IsReduced, List.isChain_flatten (by simp)]
refine ⟨⟨by simpa [IsReduced] using h.isReduced, List.isChain_replicate_of_rel _ h.2⟩,
fun _ ha _ hb ↦ ?_⟩
rw [Option.mem_def, List.getLast?_flatten_replicate (h := by simp +arith)] at ha
rw [Option.mem_def, List.head?_flatten_rep... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 75,
"column": 4
} | {
"line": 80,
"column": 23
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\nn✝ n : ℕ\nhead : α × Bool\ntail : List (α × Bool)\nh : IsCyclicallyReduced (head :: tail)\n⊢ IsCyclicallyReduced (replicate (n + 1) (head :: tail)).flatten",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"List.head?",
"Eq.mpr",
"F... | [] | rw [isCyclicallyReduced_iff, IsReduced, List.isChain_flatten (by simp)]
refine ⟨⟨by simpa [IsReduced] using h.isReduced, List.isChain_replicate_of_rel _ h.2⟩,
fun _ ha _ hb ↦ ?_⟩
rw [Option.mem_def, List.getLast?_flatten_replicate (h := by simp +arith)] at ha
rw [Option.mem_def, List.head?_flatten_rep... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 214,
"column": 8
} | {
"line": 214,
"column": 25
} | {
"line": 214,
"column": 26
} | [
{
"pp": "G : 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.18\ninst✝ : Monoid X\nf : End (root' T) →* X\na : G\n⊢ f (treeHom T a ≫ 𝟙 a ≫ inv (treeHom T a)) = 𝟙 ()",
... | [
"G : 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.18\ninst✝ : Monoid X\nf : End (root' T) →* X\na : G\n⊢ f (treeHom T a ≫ inv (treeHom T a)) = 𝟙 ()"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 665,
"column": 2
} | {
"line": 665,
"column": 40
} | {
"line": 667,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn : ℕ\n⊢ H.lowerCentralSeries n ≤ ⊤.lowerCentralSeries n",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"CompleteLattice.toBoundedOrder",
"Subgroup",
"le_top",... | [] | exact lowerCentralSeries_mono n le_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Nilpotent | {
"line": 672,
"column": 2
} | {
"line": 672,
"column": 40
} | {
"line": 674,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nK : Type u_2\ninst✝ : Group K\nf : G →* K\nn : ℕ\n⊢ (Subgroup.map f ⊤).lowerCentralSeries n ≤ ⊤.lowerCentralSeries n",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Subgroup.map",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [] | exact lowerCentralSeries_mono n le_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Nilpotent | {
"line": 960,
"column": 2
} | {
"line": 960,
"column": 81
} | {
"line": 961,
"column": 2
} | [
{
"pp": "G : Type u_2\ninst✝ : CommGroup G\n⊢ nilpotencyClass G ≤ 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommGroup.isNilpotent",
"Subgroup.upperCentralSeries",
"id",
"Subgroup",
"instOfNatNat",
"Subgroup.cente... | [
"G : Type u_2\ninst✝ : CommGroup G\n⊢ center G = ⊤"
] | rw [← upperCentralSeries_eq_top_iff_nilpotencyClass_le, upperCentralSeries_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 176,
"column": 6
} | {
"line": 176,
"column": 25
} | {
"line": 177,
"column": 4
} | [
{
"pp": "case inl\nG : 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\nH : orbit G a = {a}\ng : G\n⊢ g • a ∈ orbit G a",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"DivInvM... | [] | exact mem_orbit a g | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 183,
"column": 8
} | {
"line": 183,
"column": 44
} | {
"line": 183,
"column": 45
} | [
{
"pp": "case inr\nG : 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\nb : X\nhb : b ∈ B\ng : G\nhg : g • b = a\n⊢ IsTrivialBlock (g • B)",
"p... | [] | exact H ⟨b, hb, hg⟩ (hB.translate g) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Perm.ConjAct | {
"line": 37,
"column": 65
} | {
"line": 40,
"column": 40
} | {
"line": 42,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nk : ConjAct (Perm α)\ng : Perm α\na : α\n⊢ a ∈ (k • g).support ↔ (ConjAct.ofConjAct k⁻¹) a ∈ g.support",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"DivInvMonoid.toInv",
... | [] | by
simp only [mem_support, ConjAct.smul_def, not_iff_not, coe_mul,
Function.comp_apply, ConjAct.ofConjAct_inv]
apply Equiv.apply_eq_iff_eq_symm_apply | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 50
} | {
"line": 375,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα4 : 4 ≤ Nat.card α\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ center ↥(alternatingGroup α)\na : α\nha : a ∈ g.support\nhab : g a ≠ a\nc d : α\nhcd : c ≠ d\nhc : c ≠ a ∧ c ≠ g a\nhd : d ≠ a ∧ d ≠ g a\nk : Perm α := swap (g a) d * ... | [] | rw [swap_apply_of_ne_of_ne hab.symm hd.1.symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 380,
"column": 2
} | {
"line": 381,
"column": 39
} | {
"line": 382,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα4 : 4 ≤ Nat.card α\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ center ↥(alternatingGroup α)\na : α\nha : a ∈ g.support\nhab : g a ≠ a\nc d : α\nhcd : c ≠ d\nhc : c ≠ a ∧ c ≠ g a\nhd : d ≠ a ∧ d ≠ g a\nk : Perm α := swap (g a) d * ... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα4 : 4 ≤ Nat.card α\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ center ↥(alternatingGroup α)\na : α\nha : a ∈ g.support\nhab : g a ≠ a\nc d : α\nhcd : c ≠ d\nhc : c ≠ a ∧ c ≠ g a\nhd : d ≠ a ∧ d ≠ g a\nk : Perm α := swap (g a) d * swap (g a) c... | suffices k • (⟨g, hg⟩ : alternatingGroup α) • a = (⟨g, hg⟩ : alternatingGroup α) • k • a by
rw [this, hka]; exact hc.right.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 476,
"column": 6
} | {
"line": 476,
"column": 21
} | {
"line": 476,
"column": 21
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nhs : IsPretransitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)\ng : M\na : α\nha : a ∈ (s ∪ g • s)ᶜ\n⊢ IsPretransitive ↥(fixingSubgroup M (s ∩ g • s)) ↥(ofFixingSubgroup M (s ∩ g • s))",
"ppTerm": "?m.75",
"assi... | [
"M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nhs : IsPretransitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)\ng : M\na : α\nha : a ∈ sᶜ ∩ (g • s)ᶜ\n⊢ IsPretransitive ↥(fixingSubgroup M (s ∩ g • s)) ↥(ofFixingSubgroup M (s ∩ g • s))"
] | Set.compl_union | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 12
} | {
"line": 139,
"column": 4
} | [
{
"pp": "case h.inl\nn : 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) ↥(ofFixingSubgro... | [
"case h.inl\nn : 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... | rw [hsa] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 34
} | {
"line": 291,
"column": 6
} | [
{
"pp": "n : ℕ\nhrec :\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = n + 1 →\n n + 2 < Nat.card α →\n IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → Finite α → IsMultiplyPreprim... | [
"case right\nn : ℕ\nhrec :\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = n + 1 →\n n + 2 < Nat.card α →\n IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → Finite α → IsMultiplyPreprim... | use a, Subtype.val ⁻¹' s, ha | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 151,
"column": 2
} | {
"line": 171,
"column": 11
} | {
"line": 173,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα : 4 ≤ Nat.card α\nt : Set α\n⊢ ∀ a ∈ t, ∀ b ∈ t, ∃ g ∈ stabilizer (↥(alternatingGroup α)) t, g • a = b",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Eq.mpr",
"instHSMul",
... | [] | intro a ha b hb
by_cases hab : a = b
· use 1
simpa
by_cases ht : 2 < t.ncard
· rw [← Set.ncard_pair hab] at ht
replace ht := Set.sdiff_nonempty_of_ncard_lt_ncard ht
obtain ⟨c, hct, hc⟩ := ht
simp only [mem_insert_iff, not_or] at hc
refine ⟨⟨swap c a * swap a b, by simp [hab, hc.1]⟩, ?_, ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 151,
"column": 2
} | {
"line": 171,
"column": 11
} | {
"line": 173,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα : 4 ≤ Nat.card α\nt : Set α\n⊢ ∀ a ∈ t, ∀ b ∈ t, ∃ g ∈ stabilizer (↥(alternatingGroup α)) t, g • a = b",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Eq.mpr",
"instHSMul",
... | [] | intro a ha b hb
by_cases hab : a = b
· use 1
simpa
by_cases ht : 2 < t.ncard
· rw [← Set.ncard_pair hab] at ht
replace ht := Set.sdiff_nonempty_of_ncard_lt_ncard ht
obtain ⟨c, hct, hc⟩ := ht
simp only [mem_insert_iff, not_or] at hc
refine ⟨⟨swap c a * swap a b, by simp [hab, hc.1]⟩, ?_, ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.IsSubnormal | {
"line": 149,
"column": 2
} | {
"line": 155,
"column": 11
} | {
"line": 157,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhN : H.IsSubnormal\nne_top : H ≠ ⊤\n⊢ ∃ K, K.Normal ∧ H ≤ K ∧ K < ⊤",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Subgroup.subgroupOf",
"Preorder.toLT",
"le_rfl",
"congrArg",
"PartialOrder.toPreorder",... | [] | induction hN with
| top => contradiction
| step H K h_le hSubn hN ih =>
obtain rfl | K_ne := eq_or_ne K ⊤
· rw [normal_subgroupOf_iff_le_normalizer h_le, top_le_iff, normalizer_eq_top_iff] at hN
exact ⟨H, hN, le_rfl, ne_top.lt_top⟩
· grind | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.GroupTheory.IsSubnormal | {
"line": 149,
"column": 2
} | {
"line": 155,
"column": 11
} | {
"line": 157,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhN : H.IsSubnormal\nne_top : H ≠ ⊤\n⊢ ∃ K, K.Normal ∧ H ≤ K ∧ K < ⊤",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Subgroup.subgroupOf",
"Preorder.toLT",
"le_rfl",
"congrArg",
"PartialOrder.toPreorder",... | [] | induction hN with
| top => contradiction
| step H K h_le hSubn hN ih =>
obtain rfl | K_ne := eq_or_ne K ⊤
· rw [normal_subgroupOf_iff_le_normalizer h_le, top_le_iff, normalizer_eq_top_iff] at hN
exact ⟨H, hN, le_rfl, ne_top.lt_top⟩
· grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.IsSubnormal | {
"line": 149,
"column": 2
} | {
"line": 155,
"column": 11
} | {
"line": 157,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhN : H.IsSubnormal\nne_top : H ≠ ⊤\n⊢ ∃ K, K.Normal ∧ H ≤ K ∧ K < ⊤",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Subgroup.subgroupOf",
"Preorder.toLT",
"le_rfl",
"congrArg",
"PartialOrder.toPreorder",... | [] | induction hN with
| top => contradiction
| step H K h_le hSubn hN ih =>
obtain rfl | K_ne := eq_or_ne K ⊤
· rw [normal_subgroupOf_iff_le_normalizer h_le, top_le_iff, normalizer_eq_top_iff] at hN
exact ⟨H, hN, le_rfl, ne_top.lt_top⟩
· grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.MonoidLocalization.UniqueFactorization | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 37
} | {
"line": 31,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoidWithZero M\ninst✝ : CommMonoidWithZero N\nS : Submonoid M\nf : S.LocalizationMap N\nm : M\nprime : Prime m\nn0 : f m ≠ 0\nnu : ¬IsUnit (f m)\n⊢ Prime (f m)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCo... | [
"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoidWithZero M\ninst✝ : CommMonoidWithZero N\nS : Submonoid M\nf : S.LocalizationMap N\nm : M\nprime : Prime m\nn0 : f m ≠ 0\nnu : ¬IsUnit (f m)\nn₁ n₂ : N\ndvd : f m ∣ n₁ * n₂\n⊢ f m ∣ n₁ ∨ f m ∣ n₂"
] | refine ⟨n0, nu, fun n₁ n₂ dvd ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.Perm.Cycle.PossibleTypes | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 39
} | {
"line": 59,
"column": 4
} | [
{
"pp": "case h.right.left\nα : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl\nhc'_lt : ∀ l... | [
"case h.right.left\nα : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := ⋯\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := ⋯\nhc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α\nl : List (Li... | apply Nodup.map (Equiv.injective _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.RegularWreathProduct | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 21
} | {
"line": 172,
"column": 0
} | [
{
"pp": "case right\nD : Type u_1\nQ : Type u_2\ninst✝⁵ : Group D\ninst✝⁴ : Group Q\nΛ : Type u_3\ninst✝³ : MulAction D Λ\ninst✝² : FaithfulSMul D Λ\ninst✝¹ : Nonempty Q\ninst✝ : Nonempty Λ\nm₁ m₂ : D ≀ᵣ Q\nh : ∀ (a : Λ) (b : Q), m₁.left (m₁.right * b) • a = m₂.left (m₂.right * b) • a ∧ m₁.right = m₂.right\na :... | [] | · exact (h a b).2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.ResiduallyFinite | {
"line": 83,
"column": 86
} | {
"line": 86,
"column": 40
} | {
"line": 88,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : ResiduallyFinite G\ng h : G\nhgh : g ≠ h\n⊢ ∃ H, ↑g ≠ ↑h",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
... | [] | by
obtain ⟨H, hH⟩ :=
exists_finiteIndexNormalSubgroup_notMem (g⁻¹ * h) fun h ↦ hgh <| eq_of_inv_mul_eq_one h
exact ⟨H, by simpa [QuotientGroup.eq]⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 471,
"column": 4
} | {
"line": 471,
"column": 46
} | {
"line": 472,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nsc : ↥g.cycleFactorsFinset → ℕ := fun c ↦ #(↑c).support\nhsc : sc = fun c ↦ #(↑c).support\nthis : Fintype.card ↥(toPermHom g).range = Fintype.card ↑{k | sc ∘ ⇑k = sc}\n⊢ ∏ x ∈ Finset.image (fun c ↦ #(↑c).support) g.cycleFactorsFinset.... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nsc : ↥g.cycleFactorsFinset → ℕ := ⋯\nhsc : sc = fun c ↦ #(↑c).support\nthis : Fintype.card ↥(toPermHom g).range = Fintype.card ↑{k | sc ∘ ⇑k = sc}\n⊢ Finset.image (fun c ↦ #(↑c).support) g.cycleFactorsFinset.attach = g.cycleType.toFinset"
] | apply Finset.prod_congr _ (fun _ _ => rfl) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.PushoutI | {
"line": 343,
"column": 4
} | {
"line": 343,
"column": 28
} | {
"line": 344,
"column": 4
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nw : Word G\ni : ι\nh : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g\nhφw : ∀ (j ... | [
"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nw : Word G\ni : ι\nh : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g\nhφw : ∀ (j : ι) (g : G ... | rw [Word.equivPair_head] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 365,
"column": 6
} | {
"line": 365,
"column": 30
} | {
"line": 366,
"column": 6
} | [
{
"pp": "case refine_1.right\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nw : Word G\ni : ι\nh : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv ... | [
"case refine_1.right\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nw : Word G\ni : ι\nh : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g\nhφ... | rw [Word.equivPair_head] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 400,
"column": 8
} | {
"line": 400,
"column": 58
} | {
"line": 401,
"column": 8
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\ni✝ : ι\np : Pair d i✝\nn : ↑↑(φ i✝).range × ↑(d.set i✝) := ⋯.equiv p.h... | [
"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\ni✝ : ι\np : Pair d i✝\nn : ↑↑(φ i✝).range × ↑(d.set i✝) := ⋯.equiv p.head\nw : Wor... | rw [Word.equivPair_symm, Word.mem_rcons_iff] at hg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 204,
"column": 50
} | {
"line": 204,
"column": 53
} | {
"line": 204,
"column": 54
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : IsCyclic G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : (Nat.card G).Coprime (Nat.card K)\nhc : ¬Nat.card G = 0\nthis : Finite G\nϕ : K →* ZMod (Nat.card G) := MulDistribMulAction.toMonoidHomZModOf... | [
"G : Type u_1\ninst✝⁴ : Group G\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : IsCyclic G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : (Nat.card G).Coprime (Nat.card K)\nhc : ¬Nat.card G = 0\nthis : Finite G\nϕ : K →* ZMod (Nat.card G) := MulDistribMulAction.toMonoidHomZModOfIsCyclic G K... | hϕ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 253,
"column": 2
} | {
"line": 266,
"column": 58
} | {
"line": 268,
"column": 0
} | [
{
"pp": "G : 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\n⊢ Subgroup.normalizer ↑P ≤ Subgroup.centralizer ↑P ∨ ↑P ≤ commutator G",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Subgr... | [] | let Q : Sylow p (Subgroup.normalizer P) := P.subtype P.le_normalizer
have : Q.Normal := P.normal_in_normalizer
have : IsCyclic Q :=
isCyclic_of_surjective _ (Subgroup.subgroupOfEquivOfLe P.le_normalizer).symm.surjective
refine (le_center_or_le_commutator Q).imp (fun h ↦ ?_) (fun h ↦ ?_)
· rw [← SetLike.coe_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 253,
"column": 2
} | {
"line": 266,
"column": 58
} | {
"line": 268,
"column": 0
} | [
{
"pp": "G : 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\n⊢ Subgroup.normalizer ↑P ≤ Subgroup.centralizer ↑P ∨ ↑P ≤ commutator G",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Subgr... | [] | let Q : Sylow p (Subgroup.normalizer P) := P.subtype P.le_normalizer
have : Q.Normal := P.normal_in_normalizer
have : IsCyclic Q :=
isCyclic_of_surjective _ (Subgroup.subgroupOfEquivOfLe P.le_normalizer).symm.surjective
refine (le_center_or_le_commutator Q).imp (fun h ↦ ?_) (fun h ↦ ?_)
· rw [← SetLike.coe_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 274,
"column": 2
} | {
"line": 275,
"column": 52
} | {
"line": 276,
"column": 2
} | [
{
"pp": "G : 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\n⊢ ¬p ∣ Nat.card ↥(commutator G) ∨ ¬p ∣ (commutator G).index",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Sylow.instSetLik... | [
"case refine_1\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\nhP : Subgroup.normalizer ↑P ≤ Subgroup.centralizer ↑P\nh : p ∣ Nat.card ↥(commutator G)\n⊢ Nat.card ↥(commutator G) ∣ (↑P).index",
"case refine_2\nG : Type u_1\ninst✝³ : Grou... | refine (normalizer_le_centralizer_or_le_commutator P).imp ?_ ?_ <;>
refine fun hP h ↦ P.not_dvd_index (h.trans ?_) | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.Defs | {
"line": 232,
"column": 24
} | {
"line": 235,
"column": 57
} | {
"line": 237,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\nh : IsZeroOrMarkovKernel κ\n⊢ IsFiniteKernel κ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.IsFiniteKernel",
"MeasureTheory.Measure",
... | [] | by
rcases eq_zero_or_isMarkovKernel κ with rfl | _h'
· infer_instance
· exact ⟨⟨1, ENNReal.one_lt_top, fun _ => prob_le_one⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Basic | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 19
} | {
"line": 71,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhf : Measurable f\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ((deterministic f hf) a) s = s.indicator (fun x ↦ 1) (f a)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mea... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhf : Measurable f\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ({ toFun := fun a ↦ Measure.dirac (f a), measurable' := ⋯ } a) s = s.indicator (fun x ↦ 1) (f a)"
] | deterministic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Basic | {
"line": 120,
"column": 6
} | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 16
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ∫⁻ (a : α), f a ∂Kernel.id a = f a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.Kernel.instFunLike",
"... | [
"α : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ∫⁻ (a : α), f a ∂Measure.dirac a = f a"
] | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Basic | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 15
} | {
"line": 124,
"column": 16
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\na : α\n⊢ ∫⁻ (a : α), f a ∂Kernel.id a = f a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.Kernel.instF... | [
"α : Type u_1\nmα : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\na : α\n⊢ ∫⁻ (a : α), f a ∂Measure.dirac a = f a"
] | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Basic | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 48
} | {
"line": 288,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α β\ns t : Set β\nκ : Kernel α β\ninst✝ : IsFiniteKernel κ\nhs : MeasurableSet s\na : α\n⊢ ((κ.restrict hs) a) Set.univ ≤ κ.bound",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α β\ns t : Set β\nκ : Kernel α β\ninst✝ : IsFiniteKernel κ\nhs : MeasurableSet s\na : α\n⊢ (κ a) (Set.univ ∩ s) ≤ κ.bound"
] | rw [restrict_apply' κ hs a MeasurableSet.univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.MeasurableLIntegral | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 32
} | {
"line": 69,
"column": 4
} | [
{
"pp": "case h_fin\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nt✝ : Set (α × β)\nhκs : ∀ (a : α), IsFiniteMeasure (κ a)\nt : Set (α × β)\nhtm : MeasurableSet t\niht : Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)\nh_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \\ ... | [] | · exact measure_ne_top _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 459,
"column": 2
} | {
"line": 461,
"column": 67
} | {
"line": 463,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nδ : Type u_4\nmδ : MeasurableSpace δ\nκ : Kernel α β\nf : β → γ\ng : β → δ\nhg : Measurable g\nhf : ¬Measurable f\n⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.map f",
"ppTerm": "?neg... | [] | · have : ¬ Measurable (fun x ↦ (f x, g x)) := by
contrapose hf; exact hf.fst
simp [map_of_not_measurable _ hf, map_of_not_measurable _ this] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.CompMap | {
"line": 39,
"column": 41
} | {
"line": 42,
"column": 50
} | {
"line": 44,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nf : β → γ\nhf : Measurable f\nκ : Kernel α β\n⊢ deterministic f hf ∘ₖ κ = κ.map f",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.... | [] | by
ext a s hs
simp_rw [map_apply' _ hf _ hs, comp_apply' _ _ _ hs, deterministic_apply' hf _ hs,
lintegral_indicator_const_comp hf hs, one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.CompMap | {
"line": 45,
"column": 46
} | {
"line": 48,
"column": 52
} | {
"line": 50,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ng : γ → α\nκ : Kernel α β\nhg : Measurable g\n⊢ κ ∘ₖ deterministic g hg = κ.comap g hg",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Probabili... | [] | by
ext a s hs
simp_rw [comap_apply' _ _ _ s, comp_apply' _ _ _ hs, deterministic_apply hg a,
lintegral_dirac' _ (Kernel.measurable_coe κ hs)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 26
} | {
"line": 209,
"column": 27
} | [
{
"pp": "β : Type u_2\nmβ : MeasurableSpace β\nμ : Measure β\ninst✝ : SFinite μ\n⊢ dirac () ⊗ₘ Kernel.const Unit μ = map (Prod.mk ()) μ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"MeasureTheory.Measure",
"ProbabilityTheory.Kernel.const.in... | [
"β : Type u_2\nmβ : MeasurableSpace β\nμ : Measure β\ninst✝ : SFinite μ\n⊢ map (Prod.mk ()) ((Kernel.const Unit μ) ()) = map (Prod.mk ()) μ"
] | dirac_unit_compProd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.Prod | {
"line": 113,
"column": 31
} | {
"line": 113,
"column": 82
} | {
"line": 113,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nf : α → β\nhf : Measurable f\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\na : α\ng : β × γ → ℝ≥0∞\nhg : Measurable g\n⊢ ∫⁻ (b : β), ∫⁻ (c : γ), g (b, c) ∂κ a ∂(deterministic f hf) a = ∫⁻ (c ... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nf : α → β\nhf : Measurable f\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\na : α\ng : β × γ → ℝ≥0∞\nhg : Measurable g\n⊢ ∫⁻ (y : γ), g (f a, y) ∂κ a = ∫⁻ (c : γ), g (f a, c) ∂κ a"
] | lintegral_deterministic' _ hg.lintegral_prod_right' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.Prod | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 36
} | {
"line": 147,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\nγ✝ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ✝ : MeasurableSpace γ✝\nγ : Type u_4\nδ : Type u_5\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nη : Kernel α γ\ninst✝ : IsZeroOrMarkovKernel η\nh : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel (... | [
"case inr\nα : Type u_1\nβ : Type u_2\nγ✝ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ✝ : MeasurableSpace γ✝\nγ : Type u_4\nδ : Type u_5\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nh✝ : IsZeroOrMarkovKernel κ\nη : Kernel α γ\ninst✝ : IsZeroOrMarkovKernel η\nh : IsMarkovKernel ... | · simp only [prod]; infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.Prod | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 36
} | {
"line": 149,
"column": 2
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\nγ✝ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ✝ : MeasurableSpace γ✝\nγ : Type u_4\nδ : Type u_5\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nh✝ : IsZeroOrMarkovKernel κ\nh : IsMarkovKernel κ\ninst✝ : IsZeroOrMarkovKernel ... | [
"case inr.inr\nα : Type u_1\nβ : Type u_2\nγ✝ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ✝ : MeasurableSpace γ✝\nγ : Type u_4\nδ : Type u_5\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nh✝ : IsZeroOrMarkovKernel κ\nη : Kernel α γ\ninst✝ : IsZeroOrMarkovKernel η\nh : IsMarkovKer... | · simp only [prod]; infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.Prod | {
"line": 223,
"column": 90
} | {
"line": 225,
"column": 5
} | {
"line": 227,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\n⊢ Kernel.id = deterministic Prod.fst ⋯ ×ₖ deterministic Prod.snd ⋯",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"inferInstance",
"id",
"Measurable.pro... | [] | by
rw [deterministic_prod_deterministic]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 55,
"column": 35
} | {
"line": 57,
"column": 33
} | {
"line": 59,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\np : β → Prop\nh : ∀ᵐ (ω : β) ∂⇑κ ∘ₘ μ, p ω\n⊢ ∀ᵐ (ω' : α) ∂μ, ∀ᵐ (ω : β) ∂κ ω', p ω",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Unit.unit... | [] | by
rw [comp_eq_comp_const_apply] at h
exact Kernel.ae_ae_of_ae_comp h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 212,
"column": 81
} | {
"line": 213,
"column": 59
} | {
"line": 214,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : MeasurableSingletonClass γ\nf : α × β → γ\nhf : Measurable f\ns : Set (β × γ)\nhs : MeasurableSet s\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\nx : α\nt : Set β := {b | (b, f (x, b... | [] | by
rw [setLIntegral_congr_fun ht this, setLIntegral_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 131,
"column": 4
} | {
"line": 133,
"column": 52
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case h₂\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝² : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\nf : β × γ → E\nh1f : AEStronglyMeas... | [] | apply hasFiniteIntegral_congr
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with _ hx using
integral_congr_ae (EventuallyEq.fun_comp hx _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 131,
"column": 4
} | {
"line": 133,
"column": 52
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case h₂\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝² : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\nf : β × γ → E\nh1f : AEStronglyMeas... | [] | apply hasFiniteIntegral_congr
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with _ hx using
integral_congr_ae (EventuallyEq.fun_comp hx _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.RadonNikodym | {
"line": 68,
"column": 10
} | {
"line": 68,
"column": 17
} | {
"line": 68,
"column": 17
} | [
{
"pp": "case refine_3\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nhμν : μ ≪ ν\nκ : Kernel α β\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsFiniteKernel κ\ns : Set (α × β)\nhs : MeasurableSet s\nx✝ : (ν ⊗ₘ κ) s < ∞\nh_key :\n ∀ (t₁ : Set α... | [
"case refine_3\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nhμν : μ ≪ ν\nκ : Kernel α β\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsFiniteKernel κ\ns : Set (α × β)\nhs : MeasurableSet s\nx✝ : (ν ⊗ₘ κ) s < ∞\nh_key :\n ∀ (t₁ : Set α) (t₂ : Set ... | ← ht_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 129,
"column": 11
} | {
"line": 129,
"column": 48
} | {
"line": 129,
"column": 49
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : NeZero μ\nhf : Integrable (fun x ↦ rexp (f x)) μ\n⊢ (μ.tilted f) Set.univ = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"instHDiv",
... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : NeZero μ\nhf : Integrable (fun x ↦ rexp (f x)) μ\n⊢ ∫⁻ (a : α) in Set.univ, ENNReal.ofReal (rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) ∂μ = 1"
] | tilted_apply' _ _ MeasurableSet.univ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 35
} | {
"line": 265,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\nf : β × γ → E\ns :... | [] | simp_rw [Kernel.restrict_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 35
} | {
"line": 265,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\nf : β × γ → E\ns :... | [] | simp_rw [Kernel.restrict_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 35
} | {
"line": 265,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\nf : β × γ → E\ns :... | [] | simp_rw [Kernel.restrict_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 62
} | {
"line": 361,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : NormedSpace ℝ E\nE' : Type u_5\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace ℝ E'\nf g : ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : NormedSpace ℝ E\nE' : Type u_5\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace ℝ E'\nf g : γ → E\nF : E... | filter_upwards [hf.ae_of_comp, hg.ae_of_comp] with _ h2f h2g | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 62
} | {
"line": 368,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : NormedSpace ℝ E\nE' : Type u_5\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace ℝ E'\nf g : ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : NormedSpace ℝ E\nE' : Type u_5\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace ℝ E'\nf g : γ → E\nF : E... | filter_upwards [hf.ae_of_comp, hg.ae_of_comp] with _ h2f h2g | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 62
} | {
"line": 375,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝ : NormedSpace ℝ E\nf g : γ → E\nF : E → ℝ≥0∞\nhf : Integrable f ((η ∘ₖ κ) a)\nhg : Integrable g ((η... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝ : NormedSpace ℝ E\nf g : γ → E\nF : E → ℝ≥0∞\nhf : Integrable f ((η ∘ₖ κ) a)\nhg : Integrable g ((η ∘ₖ κ) a)\na... | filter_upwards [hf.ae_of_comp, hg.ae_of_comp] with _ h2f h2g | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.InformationTheory.KullbackLeibler.KLFun | {
"line": 188,
"column": 4
} | {
"line": 189,
"column": 44
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case hf\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\n⊢ Integrable (fun x ↦ (μ.rnDeriv ν x).toReal * log (μ.rnDeriv ν x).toReal + 1) ν",
"ppTerm": "?hf✝",
"assigned": true,
"usedCon... | [] | refine Integrable.add ?_ (integrable_const _)
rwa [integrable_rnDeriv_mul_log_iff hμν] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.InformationTheory.KullbackLeibler.KLFun | {
"line": 188,
"column": 4
} | {
"line": 189,
"column": 44
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case hf\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\n⊢ Integrable (fun x ↦ (μ.rnDeriv ν x).toReal * log (μ.rnDeriv ν x).toReal + 1) ν",
"ppTerm": "?hf✝",
"assigned": true,
"usedCon... | [] | refine Integrable.add ?_ (integrable_const _)
rwa [integrable_rnDeriv_mul_log_iff hμν] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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