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.Probability.Kernel.Defs | {
"line": 356,
"column": 16
} | {
"line": 356,
"column": 33
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\nh : IsFiniteKernel κ\na : α\ns : Set β\nhs : MeasurableSet s\ni : ℕ\n⊢ ((if i = 0 then κ else 0) a) s = if i = 0 then (κ a) s else 0",
"usedConstants": [
"Eq.mpr",
"Measur... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.Basic | {
"line": 374,
"column": 24
} | {
"line": 374,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\nt : Set β\n⊢ (if a ∈ s then κ a else η a) t = if a ∈ s then (κ a) t else (η a) t",
"usedConstants": [
"Eq.mpr",
"Mea... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.Basic | {
"line": 396,
"column": 29
} | {
"line": 396,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → ℝ≥0∞\n⊢ (∫⁻ (b : β), g b ∂if a ∈ s then κ a else η a) = if a ∈ s then ∫⁻ (b : β), g b ∂κ a else ∫⁻ (b : β), g b ∂η a",
"... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.Basic | {
"line": 401,
"column": 29
} | {
"line": 401,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → ℝ≥0∞\nt : Set β\n⊢ (∫⁻ (b : β) in t, g b ∂if a ∈ s then κ a else η a) =\n if a ∈ s then ∫⁻ (b : β) in t, g b ∂κ a else ∫⁻... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.Basic | {
"line": 408,
"column": 2
} | {
"line": 424,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)\nh' : μ ≠ 0\n⊢ ∃ η, ⇑κ =ᶠ[ae μ] ⇑η ∧ IsMarkovKernel η",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.ae",
... | classical
obtain ⟨s, s_meas, μs, hs⟩ : ∃ s, MeasurableSet s ∧ μ s = 0
∧ ∀ a ∉ s, IsProbabilityMeasure (κ a) := by
refine ⟨toMeasurable μ {a | ¬ IsProbabilityMeasure (κ a)}, measurableSet_toMeasurable _ _,
by simpa [measure_toMeasurable] using h, ?_⟩
intro a ha
contrapose! ha
exact subset_t... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Probability.Kernel.Basic | {
"line": 408,
"column": 2
} | {
"line": 424,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)\nh' : μ ≠ 0\n⊢ ∃ η, ⇑κ =ᶠ[ae μ] ⇑η ∧ IsMarkovKernel η",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.ae",
... | classical
obtain ⟨s, s_meas, μs, hs⟩ : ∃ s, MeasurableSet s ∧ μ s = 0
∧ ∀ a ∉ s, IsProbabilityMeasure (κ a) := by
refine ⟨toMeasurable μ {a | ¬ IsProbabilityMeasure (κ a)}, measurableSet_toMeasurable _ _,
by simpa [measure_toMeasurable] using h, ?_⟩
intro a ha
contrapose! ha
exact subset_t... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Basic | {
"line": 408,
"column": 2
} | {
"line": 424,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)\nh' : μ ≠ 0\n⊢ ∃ η, ⇑κ =ᶠ[ae μ] ⇑η ∧ IsMarkovKernel η",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.ae",
... | classical
obtain ⟨s, s_meas, μs, hs⟩ : ∃ s, MeasurableSet s ∧ μ s = 0
∧ ∀ a ∉ s, IsProbabilityMeasure (κ a) := by
refine ⟨toMeasurable μ {a | ¬ IsProbabilityMeasure (κ a)}, measurableSet_toMeasurable _ _,
by simpa [measure_toMeasurable] using h, ?_⟩
intro a ha
contrapose! ha
exact subset_t... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 162,
"column": 18
} | {
"line": 162,
"column": 29
} | [
{
"pp": "case h.h\nα : Type u_1\nmα : MeasurableSpace α\na : α\ns : Set (α × α)\nhs : MeasurableSet s\n⊢ (((copy α) a).bind ⇑(swap α α)) s = ((copy α) a) s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.Kernel.copy_apply",
"ProbabilityThe... | copy_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 53
} | [
{
"pp": "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\nf : β → γ\ng : γ → α\nκ : Kernel α β\ninst✝ : IsZeroOrMarkovKernel κ\nhg : Measurable g\nh : IsMarko... | have := IsMarkovKernel.comap κ hg; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 53
} | [
{
"pp": "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\nf : β → γ\ng : γ → α\nκ : Kernel α β\ninst✝ : IsZeroOrMarkovKernel κ\nhg : Measurable g\nh : IsMarko... | have := IsMarkovKernel.comap κ hg; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 92,
"column": 28
} | {
"line": 92,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nhs : MeasurableSet s\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\n⊢ (((copy α) a).bind\n ⇑(swap γ β ∘ₖ (η ∥ₖ ... | copy_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 102,
"column": 55
} | {
"line": 102,
"column": 66
} | [
{
"pp": "case e_f.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nhs : MeasurableSet s\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nb : β\n⊢ (((Measure.dirac ((a, a).1, b... | copy_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 18
} | [
{
"pp": "α : 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) γ\na : α\nb : β\ninst✝ : NeZero (κ (a, b))\n⊢ NeZero ((κ.sectR a) b)",
"usedConstants":... | rw [sectR_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Composition.Prod | {
"line": 79,
"column": 29
} | {
"line": 79,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel α γ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ (((copy α) a).bind ⇑(κ ∥ₖ η)) s = ∫⁻ (b : β), (η a) (P... | copy_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 80
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\na : ℝ≥0∞\ninst✝¹ : SFinite μ\ninst✝ : IsSFiniteKernel κ\n⊢ (a • μ) ⊗ₘ κ = a • μ ⊗ₘ κ",
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"MeasureTheory.lintegral_smul_... | ext s hs
simp only [compProd_apply hs, lintegral_smul_measure, smul_apply, smul_eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 80
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\na : ℝ≥0∞\ninst✝¹ : SFinite μ\ninst✝ : IsSFiniteKernel κ\n⊢ (a • μ) ⊗ₘ κ = a • μ ⊗ₘ κ",
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"MeasureTheory.lintegral_smul_... | ext s hs
simp only [compProd_apply hs, lintegral_smul_measure, smul_apply, smul_eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.Prod | {
"line": 161,
"column": 18
} | {
"line": 161,
"column": 29
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel α γ\ninst✝ : IsMarkovKernel η\na : α\n⊢ ((copy α) a).bind ⇑(κ ∥ₖ η).fst = κ a",
"usedConstants": [
"Eq.mpr",
"... | copy_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh2s : ((κ ⊗ₖ η) a) s ≠ ∞\nt : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s... | exact ae_lt_top (Kernel.measurable_kernel_prodMk_left' ht a) h2t | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Composition.KernelLemmas | {
"line": 113,
"column": 23
} | {
"line": 113,
"column": 68
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nT : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nmT : MeasurableSpace T\nκ : Kernel X Y\nX' : Type u_5\nmX' : MeasurableSpace X'\nη : Kernel X' Z\ninst✝¹ : IsSFiniteKernel η\nξ : Kernel Z T\ninst✝ : IsSFiniteKernel ξ\nthis :... | by simp_rw [← comp_assoc, swap_swap, id_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 33
} | [
{
"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 ∘ₖ Kernel.const Unit μ) (), p ω\n⊢ ∀ᵐ (ω' : α) ∂μ, ∀ᵐ (ω : β) ∂κ ω', p ω",
"usedConstants": [
"Unit.unit",
"ProbabilityTheory.Kernel.ae_ae_of_ae_co... | exact Kernel.ae_ae_of_ae_comp h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 63,
"column": 82
} | {
"line": 65,
"column": 23
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nh1 : μ ≪ ν\nh2 : Integrable (llr μ ν) μ\n⊢ klDiv μ ν = ENNReal.ofReal (∫ (x : α), llr μ ν x ∂μ + ν.real univ - μ.real univ)",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing... | by
rw [klDiv_def]
exact if_pos ⟨h1, h2⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 219,
"column": 49
} | {
"line": 219,
"column": 57
} | [
{
"pp": "case neg\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\nc : ℝ≥0\nhc : ¬c = 0\nh_llr_left : llr (c⁻¹ • μ) ν =ᶠ[ae μ] fun x ↦ llr μ ν x + log ↑c⁻¹\nh_llr_right : llr μ (c • ν) =ᶠ[ae μ] fun x ↦ llr... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 168,
"column": 4
} | {
"line": 172,
"column": 48
} | [
{
"pp": "case refine_2\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nf g : α → ℝ\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s μ\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s μ\nhgf : ∀ (t : Set α), MeasurableSet t → μ t < ∞ → ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, ... | rw [Measure.restrict_restrict (hm _ h_meas_nonpos_g), Measure.restrict_restrict h_meas_nonpos_f,
hgf _ (@MeasurableSet.inter α m _ _ h_meas_nonpos_g hs)
((measure_mono Set.inter_subset_right).trans_lt (lt_top_iff_ne_top.mpr hμs)),
← Measure.restrict_restrict (hm _ h_meas_nonpos_g), ←
Measure.r... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 363,
"column": 22
} | {
"line": 363,
"column": 30
} | [
{
"pp": "case neg\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\nhμ : ¬μ = 0\nhν : ¬ν = 0\nthis : ν.real univ * (μ.real univ / ν.real univ) = μ.real univ\n⊢ μ.real univ * log (μ.real univ / ν.real univ) ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 195,
"column": 8
} | {
"line": 195,
"column": 45
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\nf g : α → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nh : f =ᶠ[ae μ] g\nhm : ¬m ≤ m₀\n⊢ μ[f | m] =ᶠ[ae μ] μ[g | m]",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
... | · simp_rw [condExp_of_not_le hm]; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 76
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ η : Kernel 𝓧 𝓨\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nh_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η\nh_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ... | have ⟨hμν_ac, hκη_ac⟩ := Measure.absolutelyContinuous_compProd_iff.mp h_ac | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 289,
"column": 10
} | {
"line": 289,
"column": 59
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm₀ : MeasurableSpace α\nμ : Measure α\nf g : α → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nhf : Integrable f μ\nhg : Integrable g μ\nm : MeasurableSpace α\nhm : m ≤ m₀\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ μ[f + g | m] =ᶠ[ae ... | simp_rw [condExp_of_not_sigmaFinite hm hμm]; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 289,
"column": 10
} | {
"line": 289,
"column": 59
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm₀ : MeasurableSpace α\nμ : Measure α\nf g : α → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nhf : Integrable f μ\nhg : Integrable g μ\nm : MeasurableSpace α\nhm : m ≤ m₀\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ μ[f + g | m] =ᶠ[ae ... | simp_rw [condExp_of_not_sigmaFinite hm hμm]; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 118,
"column": 2
} | {
"line": 147,
"column": 16
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ η : Kernel 𝓧 𝓨\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nh_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η\n⊢ Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ) ↔ In... | have ⟨h_ac_μν, h_ac_κη⟩ := Measure.absolutelyContinuous_compProd_iff.mp h_ac
rw [← integrable_rnDeriv_mul_log_iff h_ac,
integrable_congr (rnDeriv_compProd_mul_log_eq_mul_add h_ac_κη),
integrable_toReal_rnDeriv_mul_iff h_ac]
have h_iff_κ : Integrable (llr μ ν) μ ↔ Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ) :=... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 118,
"column": 2
} | {
"line": 147,
"column": 16
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ η : Kernel 𝓧 𝓨\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nh_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η\n⊢ Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ) ↔ In... | have ⟨h_ac_μν, h_ac_κη⟩ := Measure.absolutelyContinuous_compProd_iff.mp h_ac
rw [← integrable_rnDeriv_mul_log_iff h_ac,
integrable_congr (rnDeriv_compProd_mul_log_eq_mul_add h_ac_κη),
integrable_toReal_rnDeriv_mul_iff h_ac]
have h_iff_κ : Integrable (llr μ ν) μ ↔ Integrable (fun x ↦ llr μ ν x.1) (μ ⊗ₘ κ) :=... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 312,
"column": 10
} | {
"line": 312,
"column": 59
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\n𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nm₀ : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nf : α → E\nm : MeasurableSpace α\nhm : m ≤ m₀\nhμm : ¬SigmaFinite (μ.trim ... | simp_rw [condExp_of_not_sigmaFinite hm hμm]; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 312,
"column": 10
} | {
"line": 312,
"column": 59
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\n𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nm₀ : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nf : α → E\nm : MeasurableSpace α\nhm : m ≤ m₀\nhμm : ¬SigmaFinite (μ.trim ... | simp_rw [condExp_of_not_sigmaFinite hm hμm]; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 461,
"column": 30
} | {
"line": 461,
"column": 67
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nfs gs : ℕ → α → E\nf g : α → E\nhfs_int : ∀ (n : ℕ), Integrable (fs n) μ\nhgs_int : ∀ (n : ℕ), Integrable (gs n) μ\nhfs : ∀ᵐ (x : α) ∂μ, Tends... | · simp_rw [condExp_of_not_le hm]; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 329,
"column": 2
} | {
"line": 332,
"column": 29
} | [
{
"pp": "α : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : InnerProductSpace 𝕜 E'\ninst✝² : CompleteSpace E'\ninst✝¹ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E'\ninst✝ : S... | refine lintegral_le_of_forall_fin_meas_trim_le hm (μ s * ‖x‖₊) fun t ht hμt => ?_
refine (setLIntegral_nnnorm_condExpL2_indicator_le hm hs hμs x ht hμt).trans ?_
gcongr
apply Set.inter_subset_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 329,
"column": 2
} | {
"line": 332,
"column": 29
} | [
{
"pp": "α : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : InnerProductSpace 𝕜 E'\ninst✝² : CompleteSpace E'\ninst✝¹ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E'\ninst✝ : S... | refine lintegral_le_of_forall_fin_meas_trim_le hm (μ s * ‖x‖₊) fun t ht hμt => ?_
refine (setLIntegral_nnnorm_condExpL2_indicator_le hm hs hμs x ht hμt).trans ?_
gcongr
apply Set.inter_subset_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 373,
"column": 63
} | {
"line": 373,
"column": 73
} | [
{
"pp": "α : Type u_1\nF : Type u_4\n𝕜 : Type u_7\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nc : 𝕜\nx : F\n⊢ (c • co... | smul_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 483,
"column": 8
} | {
"line": 483,
"column": 45
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\nf g : α → E\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : PartialOrder E\ninst✝² : ClosedIciTopology E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nhf : Integrab... | · simp_rw [condExp_of_not_le hm]; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 198,
"column": 2
} | {
"line": 206,
"column": 42
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nx : G\n⊢ ‖condExpIndL1 hm μ s x‖ ≤ μ.real s * ‖x‖",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.n... | by_cases hs : MeasurableSet s
swap
· simp_rw [condExpIndL1_of_not_measurableSet hs]; rw [Lp.norm_zero]
exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)
by_cases hμs : μ s = ∞
· rw [condExpIndL1_of_measure_eq_top hμs x, Lp.norm_zero]
exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)
· rw [co... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 198,
"column": 2
} | {
"line": 206,
"column": 42
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nx : G\n⊢ ‖condExpIndL1 hm μ s x‖ ≤ μ.real s * ‖x‖",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.n... | by_cases hs : MeasurableSet s
swap
· simp_rw [condExpIndL1_of_not_measurableSet hs]; rw [Lp.norm_zero]
exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)
by_cases hμs : μ s = ∞
· rw [condExpIndL1_of_measure_eq_top hμs x, Lp.norm_zero]
exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)
· rw [co... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 25
} | [
{
"pp": "case mpr\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : AffineSpace V P\ninst✝⁴ : Ring k\ninst✝³ : Module k V\nb : AffineBasis ι k P\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι → P\n⊢ AffineIndependent k p ∧ affineSpan k (range p) = ⊤... | rintro ⟨h_tot, h_ind⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 169,
"column": 6
} | {
"line": 170,
"column": 42
} | [
{
"pp": "case refine_1.H.e_f.h.h\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoi... | simp only [Quotient.liftOn'_mk'', coe_add, coe_smul, MultilinearMap.smul_apply,
← MultilinearMap.domCoprod'_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 169,
"column": 6
} | {
"line": 170,
"column": 42
} | [
{
"pp": "case refine_2.H.e_f.h.h\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoi... | simp only [Quotient.liftOn'_mk'', coe_add, coe_smul, MultilinearMap.smul_apply,
← MultilinearMap.domCoprod'_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 169,
"column": 6
} | {
"line": 170,
"column": 42
} | [
{
"pp": "case refine_3.H.e_f.h.h\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoi... | simp only [Quotient.liftOn'_mk'', coe_add, coe_smul, MultilinearMap.smul_apply,
← MultilinearMap.domCoprod'_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 169,
"column": 6
} | {
"line": 170,
"column": 42
} | [
{
"pp": "case refine_4.H.e_f.h.h\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoi... | simp only [Quotient.liftOn'_mk'', coe_add, coe_smul, MultilinearMap.smul_apply,
← MultilinearMap.domCoprod'_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nP : Type u_5\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : Module S P\ninst✝ : IsScalarTower R S P\nα : M →ₗ[R] P\nj : IsBaseChange... | simp [endHom_apply, IsBaseChange.equiv_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nP : Type u_5\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : Module S P\ninst✝ : IsScalarTower R S P\nα : M →ₗ[R] P\nj : IsBaseChange... | simp [endHom_apply, IsBaseChange.equiv_symm_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nP : Type u_5\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : Module S P\ninst✝ : IsScalarTower R S P\nα : M →ₗ[R] P\nj : IsBaseChange... | simp [endHom_apply, IsBaseChange.equiv_symm_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Center | {
"line": 149,
"column": 12
} | {
"line": 149,
"column": 20
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Center | {
"line": 155,
"column": 12
} | {
"line": 155,
"column": 20
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 113,
"column": 77
} | {
"line": 113,
"column": 86
} | [
{
"pp": "R : Type u_3\nV : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid V\ninst✝³ : Module R V\ninst✝² : Free R V\ninst✝¹ : Module.Finite R V\ninst✝ : StrongRankCondition R\nf : Dual R V\nv : V\nh : finrank R V = 1\nh1 : 1 ≤ 1\nb : Basis (Fin (finrank R V)) R V := finBasis R V\nx : V\ni : Fin (finr... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 94,
"column": 30
} | {
"line": 94,
"column": 38
} | [
{
"pp": "case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' d₁ d₂ : Module.Dual R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx :\n (foldr' Q (contractLeftAux Q (d₁ + d₂)) ⋯ 0) x✝ =\n (foldr' Q (contractLeftAux Q d₁) ⋯ 0) x✝ + (foldr' ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 104,
"column": 34
} | {
"line": 104,
"column": 45
} | [
{
"pp": "case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd✝ d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (foldr' Q (contractLeftAux Q (c • d)) ⋯ 0) x✝ = c • (foldr' Q (contractLeftAux Q d) ⋯ ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 158,
"column": 42
} | {
"line": 158,
"column": 51
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : M\n⊢ d x • 1 - (ι Q) x * 0 = (algebraMap R (CliffordAlgebra Q)) (d x)",
"usedConstants": [
"Eq.mpr",
"CliffordAlgebra.ι",
"instHSMul",
... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 192,
"column": 71
} | {
"line": 192,
"column": 80
} | [
{
"pp": "case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (contractLeft d) ((contractLeft d) x✝) = 0\n⊢ d m✝ • (contractLeft d) x✝ - (d m✝ • (contractLeft d) x✝ - (ι Q) m✝ * 0)... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 86,
"column": 31
} | {
"line": 86,
"column": 39
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : CommRing R\ny x₁ x₂ : CliffordAlgebra 0\nhx₁ : x₁ * y = y * x₁\nhx₂ : x₂ * y = y * x₂\n⊢ (x₁ + x₂) * y = y * (x₁ + x₂)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModu... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Even | {
"line": 185,
"column": 41
} | {
"line": 185,
"column": 55
} | [
{
"pp": "case snd.a.h.refine_1\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : EvenHom Q A\nm : M\na : A\ng : M →ₗ[R] A\nhg : g ∈ S f\nm₁ : M\nb : A\nm₃ : M\n⊢ (f.bilin m₁) m * (f.bilin... | f.contract_mid | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Even | {
"line": 188,
"column": 52
} | {
"line": 188,
"column": 60
} | [
{
"pp": "case snd.a.h.refine_3\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : EvenHom Q A\nm : M\na : A\ng : M →ₗ[R] A\nhg : g ∈ S f\nm₁ : M\nx y : M →ₗ[R] A\n_hx : x ∈ Submodule.span ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 329,
"column": 6
} | {
"line": 329,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc₁ c₂ : R\nq : ℍ[R,c₁,c₂]\n⊢ star q = star (toQuaternion (ofQuaternion q))",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"CommRing.toNonUnitalCommRing",
"CliffordAlgebraQuaternion.... | toQuaternion_ofQuaternion | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv | {
"line": 191,
"column": 60
} | {
"line": 191,
"column": 68
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nm₁✝ m₂✝ : M × R\nm₁ : M\nr₁ : R\nm₂ : M\nr₂ : R\n⊢ (v Q) m₁ * (v Q) m₂ + r₁ • e0 Q * (v Q) m₂ + (v Q) m₁ * r₂ • e0 Q + r₁ • e0 Q * r₂ • e0 Q =\n ((v Q) m₁ + r₁ • e0 Q) * ((v Q) m₂ + r... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 59,
"column": 36
} | {
"line": 59,
"column": 45
} | [
{
"pp": "case zero\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : QuadraticForm R N... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 60,
"column": 49
} | {
"line": 60,
"column": 57
} | [
{
"pp": "case add\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : QuadraticForm R N\... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 66,
"column": 42
} | {
"line": 66,
"column": 51
} | [
{
"pp": "case mem.algebraMap\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : Quadrat... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 68,
"column": 28
} | {
"line": 68,
"column": 36
} | [
{
"pp": "case mem.add\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : QuadraticForm ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 72,
"column": 63
} | {
"line": 72,
"column": 77
} | [
{
"pp": "case mem.mem_mul\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : QuadraticF... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 77,
"column": 40
} | {
"line": 77,
"column": 49
} | [
{
"pp": "case mem.mem_mul.e_a.e_a.zero\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 142,
"column": 29
} | {
"line": 142,
"column": 38
} | [
{
"pp": "case H.h\nR : Type u_1\nι : Type u_2\ninst✝⁷ : CommSemiring ι\ninst✝⁶ : Module ι (Additive ℤˣ)\ninst✝⁵ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝² : (i : ι) → AddCommGroup (ℬ i)\ninst✝¹ : (i : ι) → Module R (𝒜 i)\ninst✝ : (i... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 78,
"column": 53
} | {
"line": 78,
"column": 61
} | [
{
"pp": "case mem.mem_mul.e_a.e_a.add\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 87,
"column": 32
} | {
"line": 87,
"column": 40
} | [
{
"pp": "case mem.mem_mul.e_a.e_a.mem.add\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 91,
"column": 36
} | {
"line": 91,
"column": 50
} | [
{
"pp": "case mem.mem_mul.e_a.e_a.mem.mem_mul\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 227,
"column": 6
} | {
"line": 227,
"column": 15
} | [
{
"pp": "case a.H.h.H.h\nR : Type u_1\nι : Type u_2\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\nin... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 76,
"column": 6
} | {
"line": 93,
"column": 77
} | [
{
"pp": "case mem.mem_mul.e_a.e_a\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : Qu... | induction hm₂ using Submodule.iSup_induction' with
| zero => rw [map_zero, zero_mul, mul_zero, smul_zero]
| add _ _ _ _ ihx ihy => rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]
| mem i₂' m₂' hm₂ =>
clear m₂
obtain ⟨i₂n, rfl⟩ := i₂'
dsimp only at *
induction hm₂ usi... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 124,
"column": 17
} | {
"line": 124,
"column": 25
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : QuadraticForm R N\nm : M₁ × ... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 270,
"column": 24
} | {
"line": 270,
"column": 32
} | [
{
"pp": "case a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R ... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.ExteriorPower.Pairing | {
"line": 119,
"column": 8
} | {
"line": 119,
"column": 23
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : LinearOrder ι\nx : ι → M\nf : ι → Module.Dual R M\nh₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0\nn : ℕ\na b : Fin n ↪o ι\nh : a ≠ b\nσ : Equiv.Perm (Fin n)\nx✝ : σ ∈ Finset.univ\nh' : ¬∏ x_... | ← a.map_rel_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 63,
"column": 2
} | {
"line": 65,
"column": 55
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ (Matrix.of fun i j ↦ (b.coord ((ofFinEmbEquiv.symm s) j)) ((⇑b ∘ ⇑(ofFinEmbEquiv.symm s)) i)).det = 1",
"usedConstan... | suffices Matrix.of (fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j)
(b (powersetCard.ofFinEmbEquiv.symm s i))) = 1 by
simp_rw [Function.comp_apply, this, Matrix.det_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 208,
"column": 29
} | {
"line": 208,
"column": 38
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgebra ℬ\... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Determinant.Misc | {
"line": 80,
"column": 21
} | {
"line": 80,
"column": 30
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin (n + 1)) (Fin (n + 1)) R\ni₀ j₀ : Fin (n + 1)\nhv : ∀ (j : Fin (n + 1)), j ≠ j₀ → ∑ i, M i j = 0\ni : { i // i ≠ j₀ }\n| (-1) ^ (↑i₀ + ↑↑i) * 0 * (M.submatrix i₀.succAbove (↑i).succAbove).det",
"usedConstants": [
"Fin.succAbove"... | mul_zero, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 155,
"column": 25
} | {
"line": 155,
"column": 59
} | [
{
"pp": "A : FixedDetMatrix (Fin 2) ℤ 0\nh₁ : ↑A 1 0 = 0\nh₂ : 0 < ↑A 0 0\nh₄ : |↑A 0 1| < |↑A 1 1|\nthis : |↑A 0 1| < 0\n⊢ False",
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithO... | by linarith [abs_nonneg (A.1 0 1)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 25
} | [
{
"pp": "case h.mp\nR : Type u_1\ninst✝² : Semiring R\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix n n R\n⊢ (∀ (i j : n), M i j = 0) → M = 0",
"usedConstants": [
"Matrix",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"NonAssocSemiring.toNonUnitalNonAssocSemiring",
... | · intro H; ext; apply H | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Permanent | {
"line": 86,
"column": 50
} | {
"line": 87,
"column": 94
} | [
{
"pp": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nσ : Perm n\nM : Matrix n n R\n⊢ (M.submatrix id ⇑σ).permanent = M.permanent",
"usedConstants": [
"Eq.mpr",
"Matrix.submatrix",
"Equiv.instEquivLike",
"Matrix.transpose_submatrix",... | by
rw [← permanent_transpose, transpose_submatrix, permanent_permute_cols, permanent_transpose] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Module | {
"line": 40,
"column": 4
} | {
"line": 41,
"column": 24
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝⁸ : Ring R\ninst✝⁷ : Fintype ι\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nN₁ N₂ : Matrix ι ι R\nv : ι → M\ni ... | simp_rw [mul_apply, Finset.smul_sum, Finset.sum_smul, SemigroupAction.mul_smul]
rw [Finset.sum_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Module | {
"line": 40,
"column": 4
} | {
"line": 41,
"column": 24
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝⁸ : Ring R\ninst✝⁷ : Fintype ι\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nN₁ N₂ : Matrix ι ι R\nv : ι → M\ni ... | simp_rw [mul_apply, Finset.smul_sum, Finset.sum_smul, SemigroupAction.mul_smul]
rw [Finset.sum_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basis | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 51
} | [
{
"pp": "case H\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nbm : Basis ι R M\nx : M\n⊢ ∑ x_1 ∈ (bm.repr x).support, (bm.repr x) x_1 • (bm.r... | ← polar_smul_left _ (bm.repr x <| Prod.fst _) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 186,
"column": 42
} | {
"line": 186,
"column": 51
} | [
{
"pp": "case neg\n𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set ι\nhs : ∀ i ∈ s, w i ≤ 0\nx : ι → 𝕜\nhx : x ∈ Pi.spanSubset 𝕜 s\ni : ι\na✝ : i ∈ univ\nhi : i ∉ s\n⊢ w i * (0 * 0) ≤ 0",
"usedConstants": [
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 179,
"column": 63
} | {
"line": 186,
"column": 61
} | [
{
"pp": "𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set ι\nhs : ∀ i ∈ s, w i ≤ 0\n⊢ ∀ x ∈ Pi.spanSubset 𝕜 s, (weightedSumSquares 𝕜 w) x ≤ 0",
"usedConstants": [
"mul_self_nonneg",
"Eq.mpr",
... | by
intro x hx
simp only [weightedSumSquares_apply, smul_eq_mul]
apply sum_nonpos
intro i _
by_cases hi : i ∈ s
· exact mul_nonpos_of_nonpos_of_nonneg (hs i hi) (mul_self_nonneg _)
· rw [Pi.mem_spanSubset_iff.mp hx i hi, mul_zero, mul_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 99
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P.Base\ninst✝ : DecidableEq ι\nd : ι → R\ng : Matrix ι ι R →ₗ[R] Matrix (↥... | have h₀ : Injective (g ∘ diagonalLinearMap ι R R) := fun _ _ hd ↦ funext <| by simpa [g] using hd | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 410,
"column": 8
} | {
"line": 410,
"column": 24
} | [
{
"pp": "case lie\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZe... | LieEquiv.map_lie | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 158,
"column": 2
} | {
"line": 161,
"column": 24
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | have hmi' : P.root m ≠ -P.root i := fun contra ↦ by
replace h₂ : P.root k = -P.root i + P.root j := by rwa [contra, sub_eq_iff_eq_add] at h₂
replace h₃ : P.root n = 0 := by rw [h₃, h₂]; abel
exact P.ne_zero _ h₃ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 85
} | [
{
"pp": "case inr.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\nb ... | suffices P.chainBotCoeff i k = 0 ∧ P.chainTopCoeff i k = 0 by simp [h₁, h₂, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 420,
"column": 41
} | {
"line": 420,
"column": 49
} | [
{
"pp": "case calc_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.EmbeddedG2\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\ni : ι\nthis :... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 451,
"column": 43
} | {
"line": 451,
"column": 51
} | [
{
"pp": "case calc_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.EmbeddedG2\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\ni : ι\nthis :... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 249,
"column": 59
} | {
"line": 249,
"column": 64
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | aux₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 251,
"column": 6
} | {
"line": 251,
"column": 37
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | rw [← P.algebraMap_pairingIn ℤ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 225,
"column": 32
} | {
"line": 225,
"column": 48
} | [
{
"pp": "case h.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P... | lie_e_f_ne_aux₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 256,
"column": 32
} | {
"line": 256,
"column": 48
} | [
{
"pp": "case h.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P... | lie_e_f_ne_aux₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 306,
"column": 6
} | {
"line": 306,
"column": 57
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb... | simp [← P.root_coroot_eq_pairing l, ← h₁, add_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 304,
"column": 6
} | {
"line": 306,
"column": 13
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Finite ι\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\nb : P... | have h₇ : P.root l - P.root j ∉ range P.root := by
rwa [b.root_sub_mem_iff_root_add_mem i j l hij' i.property j.property h₃ _ h₅]
simpa | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.OfBilinear | {
"line": 61,
"column": 38
} | {
"line": 61,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : M →ₗ[R] M →ₗ[R] R\nx : M\nhx : B.IsReflective x\na b : M\n⊢ 2 * (B x) (a + b) = (B x) x * (Exists.choose ⋯ + Exists.choose ⋯)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Decomposition | {
"line": 47,
"column": 5
} | {
"line": 47,
"column": 67
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nS : Type u_4\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nℳ : ι → Submodule R M\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : Decomposition ℳ\ni : ι\nx y : S ⊗[R] ↥(ℳ i)\nh : (Submodule.toBaseChange ... | by rw [← LinearEquiv.coe_trans]; exact LinearEquiv.bijective _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 41
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Field K\ninst✝⁸ : CharZero K\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module K M\ninst✝³ : AddCommGroup N\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallog... | ← iSup_genWeightSpace_eq_top K H U, | Lean.Elab.Tactic.evalRewriteSeq | null |
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