module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Kernel.Basic | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 46
} | {
"line": 398,
"column": 0
} | [
{
"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 ∂(piecewise hs κ η) a = if a ∈ s then ∫⁻ (b : β), g b ∂κ a else ∫⁻ (b : β), g b ∂η a",
"ppTerm":... | [] | simp_rw [piecewise_apply]; split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Basic | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 46
} | {
"line": 403,
"column": 0
} | [
{
"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 ∂(piecewise hs κ η) a = if a ∈ s then ∫⁻ (b : β) in t, g b ∂κ a else ∫⁻ (b : β) in t... | [] | simp_rw [piecewise_apply]; split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Basic | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 46
} | {
"line": 403,
"column": 0
} | [
{
"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 ∂(piecewise hs κ η) a = if a ∈ s then ∫⁻ (b : β) in t, g b ∂κ a else ∫⁻ (b : β) in t... | [] | simp_rw [piecewise_apply]; split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Basic | {
"line": 413,
"column": 12
} | {
"line": 413,
"column": 14
} | {
"line": 414,
"column": 4
} | [
{
"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\na : α\n⊢ a ∉ toMeasurable μ {a | ¬IsProbabilityMeasure (κ a)} → IsProbabilityMeasure (κ a)",
"ppTerm": "?m.79",
"assigned": true,... | [
"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\na : α\nha : a ∉ toMeasurable μ {a | ¬IsProbabilityMeasure (κ a)}\n⊢ IsProbabilityMeasure (κ a)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 131,
"column": 75
} | {
"line": 131,
"column": 77
} | {
"line": 131,
"column": 78
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\np : α → Prop\nhp : MeasurableSet {x | p x}\nh : ∀ᵐ (a : α) ∂μ, p a\na : α\n⊢ p a → ∀ᵐ (b : β) ∂κ a, p a",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\np : α → Prop\nhp : MeasurableSet {x | p x}\nh : ∀ᵐ (a : α) ∂μ, p a\na : α\nha : p a\n⊢ ∀ᵐ (b : β) ∂κ a, p a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Composition.CompMap | {
"line": 51,
"column": 89
} | {
"line": 52,
"column": 75
} | {
"line": 54,
"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\ng : γ → α\nhf : Measurable f\nhg : Measurable g\n⊢ deterministic g hg ∘ₖ deterministic f hf = deterministic (g ∘ f) ⋯",
"ppTerm": "?m.39",
"assigned": true,
"usedCons... | [] | by
ext; simp [comp_deterministic_eq_comap, comap_apply, deterministic_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 412,
"column": 66
} | {
"line": 412,
"column": 79
} | {
"line": 414,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α (β × γ)\n⊢ κ.fst = κ.map Prod.fst",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"ProbabilityTheory.Kernel.map",
"Prod.fst"... | [] | by simp [fst] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 425,
"column": 51
} | {
"line": 425,
"column": 64
} | {
"line": 427,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\n⊢ fst 0 = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"congrArg",
"ProbabilityTheory.Kernel.map",
"Prod.fst",
"ProbabilityTheory.Kernel.m... | [] | by simp [fst] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 285,
"column": 2
} | {
"line": 301,
"column": 22
} | {
"line": 303,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ η : Kernel α β\ninst✝¹ : SFinite μ\ninst✝ : IsSFiniteKernel κ\nh_zero : ∀ (a : α), NeZero (κ a)\nh : μ ⊗ₘ κ ≪ ν ⊗ₘ η\n⊢ μ ≪ ν",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Measu... | [] | refine Measure.AbsolutelyContinuous.mk (fun s hs hs0 ↦ ?_)
have h1 : (ν ⊗ₘ η) (s ×ˢ univ) = 0 := by
by_cases hν : SFinite ν
swap; · simp [compProd_of_not_sfinite _ _ hν]
by_cases hη : IsSFiniteKernel η
swap; · simp [compProd_of_not_isSFiniteKernel _ _ hη]
rw [Measure.compProd_apply_prod hs Measura... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 285,
"column": 2
} | {
"line": 301,
"column": 22
} | {
"line": 303,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ η : Kernel α β\ninst✝¹ : SFinite μ\ninst✝ : IsSFiniteKernel κ\nh_zero : ∀ (a : α), NeZero (κ a)\nh : μ ⊗ₘ κ ≪ ν ⊗ₘ η\n⊢ μ ≪ ν",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Measu... | [] | refine Measure.AbsolutelyContinuous.mk (fun s hs hs0 ↦ ?_)
have h1 : (ν ⊗ₘ η) (s ×ˢ univ) = 0 := by
by_cases hν : SFinite ν
swap; · simp [compProd_of_not_sfinite _ _ hν]
by_cases hη : IsSFiniteKernel η
swap; · simp [compProd_of_not_isSFiniteKernel _ _ hη]
rw [Measure.compProd_apply_prod hs Measura... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 89
} | {
"line": 205,
"column": 2
} | [
{
"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... | [
"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)) ∈ s}\nht ... | have ht : MeasurableSet t := (measurable_id.prodMk (hf.comp measurable_prodMk_left)) hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 383,
"column": 2
} | {
"line": 383,
"column": 46
} | {
"line": 384,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ η : Kernel α β\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\n⊢ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν.withDensity (μ.rnDeriv ν) ⊗ₘ η ↔\n ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η",
... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ η : Kernel α β\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nh : ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν.withDensity (μ.rnDeriv ν) ⊗ₘ η\nξ : Measure α\nhξ : SFinite ξ\nhξμ : ξ ≪ μ\nhξν : ξ ≪ ν\n⊢ ξ ⊗ₘ κ ⟂ₘ... | refine ⟨fun h ξ hξ hξμ hξν ↦ ?_, fun h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 100
} | {
"line": 185,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\ninst✝¹ : Countable α\ninst✝ : MeasurableSingletonClass α\n⊢ ∀ᵐ (ω : α) ∂μ, κ ω ≪ ⇑κ ∘ₘ μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.m... | [] | rw [Measure.comp_eq_sum_of_countable, ae_iff_of_countable]
exact fun ω hμω ↦ Measure.absolutelyContinuous_sum_right ω (Measure.absolutelyContinuous_smul hμω) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 100
} | {
"line": 185,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\ninst✝¹ : Countable α\ninst✝ : MeasurableSingletonClass α\n⊢ ∀ᵐ (ω : α) ∂μ, κ ω ≪ ⇑κ ∘ₘ μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.m... | [] | rw [Measure.comp_eq_sum_of_countable, ae_iff_of_countable]
exact fun ω hμω ↦ Measure.absolutelyContinuous_sum_right ω (Measure.absolutelyContinuous_smul hμω) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 386,
"column": 61
} | {
"line": 386,
"column": 63
} | {
"line": 386,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nf : β × γ → ℝ≥0∞\nhf : AEMeasurable f ((κ ⊗ₖ η) a)\nA : ∫⁻ (z : β × γ), f z ∂(κ ⊗ₖ η) a =... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nf : β × γ → ℝ≥0∞\nhf : AEMeasurable f ((κ ⊗ₖ η) a)\nA : ∫⁻ (z : β × γ), f z ∂(κ ⊗ₖ η) a = ∫⁻ (z : β ×... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 97
} | {
"line": 119,
"column": 2
} | [
{
"pp": "𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure 𝓧\n𝓨 : Type u_2\nm𝓨 : MeasurableSpace 𝓨\nκ η : Kernel 𝓧 𝓨\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsFiniteKernel η\nhκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η\n⊢ ∀ᵐ (a : 𝓧) ∂μ, ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂η a = (κ a) uni... | [
"𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure 𝓧\n𝓨 : Type u_2\nm𝓨 : MeasurableSpace 𝓨\nκ η : Kernel 𝓧 𝓨\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsFiniteKernel η\nhκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η\n⊢ ∀ (s : Set 𝓧),\n MeasurableSet s →\n μ s < ∞ → ∫⁻ (x : 𝓧) in s, ∫⁻ (b : 𝓨), (μ ... | refine ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite (by fun_prop) (κ.measurable_coe .univ) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv | {
"line": 148,
"column": 13
} | {
"line": 148,
"column": 15
} | {
"line": 149,
"column": 4
} | [
{
"pp": "𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\n𝓨 : Type u_2\nm𝓨 : MeasurableSpace 𝓨\nκ η : Kernel 𝓧 𝓨\nf : ℝ → ℝ\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nhf : StronglyMeasurable f\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_co... | [
"𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\n𝓨 : Type u_2\nm𝓨 : MeasurableSpace 𝓨\nκ η : Kernel 𝓧 𝓨\nf : ℝ → ℝ\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nhf : StronglyMeasurable f\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_cont_at : Cont... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 42,
"column": 41
} | {
"line": 42,
"column": 43
} | {
"line": 42,
"column": 44
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\na : α\n⊢ μ.rnDeriv μ a = 1 → llr μ μ a = 0 a",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.rnDeriv",
"ENNReal",
"One.toOfNat1",
"ENNReal.instOne",
"Of... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\na : α\nha : μ.rnDeriv μ a = 1\n⊢ llr μ μ a = 0 a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 344,
"column": 88
} | {
"line": 348,
"column": 43
} | {
"line": 350,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhfν : AEMeasurable f ν\n⊢ (μ.tilted f).rnDeriv ν =ᵐ[ν] fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * μ.rnDeriv ν x",
"ppTerm": "?m.38",
"assigned": tr... | [] | by
let g := fun x ↦ ENNReal.ofReal (exp (f x) / (∫ x, exp (f x) ∂μ))
refine Measure.rnDeriv_withDensity_left (μ := μ) (ν := ν) (f := g) ?_ ?_
· exact ((measurable_exp.comp_aemeasurable hfν).div_const _).ennreal_ofReal
· exact ae_of_all _ (fun x ↦ by simp [g]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 317,
"column": 82
} | {
"line": 317,
"column": 84
} | {
"line": 317,
"column": 84
} | [
{
"pp": "α : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E'\nt : Set α\... | [
"α : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E'\nt : Set α\nht : Measur... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 385,
"column": 75
} | {
"line": 385,
"column": 77
} | {
"line": 385,
"column": 77
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : G\nt : Set α\nht : MeasurableSet t\nhμt : μ t ≠ ∞\na : α\nha : ↑↑(condExpIndSMul hm hs hμs x) a = (fun a ↦ ↑↑↑((c... | [
"α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : G\nt : Set α\nht : MeasurableSet t\nhμt : μ t ≠ ∞\na : α\nha : ↑↑(condExpIndSMul hm hs hμs x) a = (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 394,
"column": 2
} | {
"line": 397,
"column": 29
} | {
"line": 399,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : G\ninst✝ : SigmaFinite (μ.trim hm)\n⊢ ∫⁻ (a : α), ↑‖↑↑(condExpIndSMul hm hs hμs x) a‖₊ ∂μ ≤ μ s * ↑‖x‖₊",
"p... | [] | refine lintegral_le_of_forall_fin_meas_trim_le hm (μ s * ‖x‖₊) fun t ht hμt => ?_
refine (setLIntegral_nnnorm_condExpIndSMul_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": 394,
"column": 2
} | {
"line": 397,
"column": 29
} | {
"line": 399,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : G\ninst✝ : SigmaFinite (μ.trim hm)\n⊢ ∫⁻ (a : α), ↑‖↑↑(condExpIndSMul hm hs hμs x) a‖₊ ∂μ ≤ μ s * ↑‖x‖₊",
"p... | [] | refine lintegral_le_of_forall_fin_meas_trim_le hm (μ s * ‖x‖₊) fun t ht hμt => ?_
refine (setLIntegral_nnnorm_condExpIndSMul_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": 464,
"column": 63
} | {
"line": 464,
"column": 65
} | {
"line": 465,
"column": 2
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nE : Type u_10\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : PartialOrder E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : IsOrderedModule ℝ E\ninst✝ : SigmaFinite (μ.trim hm)\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E\nhx : 0 ≤ x\na : α\n⊢... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nE : Type u_10\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : PartialOrder E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : IsOrderedModule ℝ E\ninst✝ : SigmaFinite (μ.trim hm)\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E\nhx : 0 ≤ x\na : α\nha : 0 a ≤ ↑↑... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 45,
"column": 33
} | {
"line": 45,
"column": 35
} | {
"line": 45,
"column": 36
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HereditarilyLindelofSpace E\nhm : m ≤ mα\nhf_int : Integrable f μ\nL : ℕ → StrongDual ℝ E\nc ... | [
"E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HereditarilyLindelofSpace E\nhm : m ≤ mα\nhf_int : Integrable f μ\nL : ℕ → StrongDual ℝ E\nc : ℕ → ℝ\nhs ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 31
} | {
"line": 516,
"column": 2
} | [
{
"pp": "α : Type u_1\nF' : Type u_3\n𝕜 : Type u_6\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace 𝕜 F'\ninst✝¹ : NormedSpace ℝ F'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nc : 𝕜\nf : α → F'\n⊢ condExpL1 hm μ (c • f) = c • condExpL1 hm μ... | [
"α : Type u_1\nF' : Type u_3\n𝕜 : Type u_6\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace 𝕜 F'\ninst✝¹ : NormedSpace ℝ F'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nc : 𝕜\nf : α → F'\n⊢ ∀ (c : 𝕜) (s : Set α) (x : F'), (condExpInd F' hm μ s)... | refine setToFun_smul _ ?_ c f | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 61,
"column": 32
} | {
"line": 61,
"column": 34
} | {
"line": 61,
"column": 35
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nt... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nthis✝² : Meas... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 64,
"column": 33
} | {
"line": 64,
"column": 35
} | {
"line": 65,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nt... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nthis✝² : Meas... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 116,
"column": 2
} | {
"line": 137,
"column": 74
} | {
"line": 139,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\n⊢ (μ.restrict s)[f | m] =ᵐ[μ.restr... | [] | have : SigmaFinite ((μ.restrict s).trim hm) := by rw [← restrict_trim hm _ hs_m]; infer_instance
rw [ae_eq_restrict_iff_indicator_ae_eq (hm _ hs_m)]
refine EventuallyEq.trans ?_ (condExp_indicator hf_int hs_m)
refine ae_eq_condExp_of_forall_setIntegral_eq hm (hf_int.indicator (hm _ hs_m)) ?_ ?_ ?_
· intro t ht ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 116,
"column": 2
} | {
"line": 137,
"column": 74
} | {
"line": 139,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\n⊢ (μ.restrict s)[f | m] =ᵐ[μ.restr... | [] | have : SigmaFinite ((μ.restrict s).trim hm) := by rw [← restrict_trim hm _ hs_m]; infer_instance
rw [ae_eq_restrict_iff_indicator_ae_eq (hm _ hs_m)]
refine EventuallyEq.trans ?_ (condExp_indicator hf_int hs_m)
refine ae_eq_condExp_of_forall_setIntegral_eq hm (hf_int.indicator (hm _ hs_m)) ?_ ?_ ?_
· intro t ht ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 67,
"column": 37
} | {
"line": 67,
"column": 39
} | {
"line": 67,
"column": 40
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nt... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nthis✝² : Meas... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 451,
"column": 31
} | {
"line": 451,
"column": 33
} | {
"line": 451,
"column": 34
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nι : Type u_5\ninst✝ : Countable ι\nf : ι → α → E\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞\nhm : m ≤... | [
"α : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nι : Type u_5\ninst✝ : Countable ι\nf : ι → α → E\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞\nhm : m ≤ m₀\nhμm : S... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 74,
"column": 32
} | {
"line": 74,
"column": 34
} | {
"line": 75,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nt... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nthis✝² : Meas... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 81,
"column": 70
} | {
"line": 81,
"column": 72
} | {
"line": 81,
"column": 73
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nt... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nthis✝² : Meas... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 81
} | {
"line": 475,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace 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, Tendsto (fun n ... | [
"α : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace 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, Tendsto (fun n ↦ fs n x) at... | refine (condExp_ae_eq_condExpL1 hm (gs n)).symm.trans ((hfg n).symm.trans ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 96,
"column": 33
} | {
"line": 96,
"column": 35
} | {
"line": 96,
"column": 36
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nn : ℕ\n... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 110,
"column": 33
} | {
"line": 110,
"column": 35
} | {
"line": 110,
"column": 36
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HereditarilyLindelofSpace E\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : Lower... | [
"E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HereditarilyLindelofSpace E\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuo... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 130,
"column": 32
} | {
"line": 130,
"column": 34
} | {
"line": 130,
"column": 35
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 133,
"column": 33
} | {
"line": 133,
"column": 35
} | {
"line": 134,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym | {
"line": 70,
"column": 10
} | {
"line": 70,
"column": 58
} | {
"line": 71,
"column": 8
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝ : IsFiniteMeasure μ\nhμν : μ ≪ ν\ng : 𝓧 → 𝓨\nhg : Measurable g\nhσ : SigmaFinite (Measure.map g ν)\nthis✝ : SigmaFinite (ν.trim ⋯)\nthis : SigmaFinite ν\nt : Set 𝓨\nht : MeasurableSet t\na✝ : ν ... | [
"𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝ : IsFiniteMeasure μ\nhμν : μ ≪ ν\ng : 𝓧 → 𝓨\nhg : Measurable g\nhσ : SigmaFinite (Measure.map g ν)\nthis✝ : SigmaFinite (ν.trim ⋯)\nthis : SigmaFinite ν\nt : Set 𝓨\nht : MeasurableSet t\na✝ : ν (g ⁻¹' t) < ... | Measure.setIntegral_toReal_rnDeriv (hμν.map hg), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner | {
"line": 79,
"column": 11
} | {
"line": 79,
"column": 13
} | {
"line": 79,
"column": 14
} | [
{
"pp": "𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ : Measure 𝓧\nf : 𝓧 → ℝ\nhf : Integrable f μ\nh'f : 0 ≤ᵐ[μ] f\nhm : m ≤ m𝓧\nhμ : SigmaFinite (μ.trim hm)\nA : μ[fun x ↦ (ENNReal.ofReal (f x)).toReal | m] =ᵐ[μ] μ[f | m]\nB : 0 ≤ᵐ[μ] μ[f | m]\ng : 𝓧 → ℝ≥0∞ := fun x ↦ ENNReal.ofReal (f x)\nI : ∫⁻ (x : 𝓧),... | [
"𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ : Measure 𝓧\nf : 𝓧 → ℝ\nhf : Integrable f μ\nh'f : 0 ≤ᵐ[μ] f\nhm : m ≤ m𝓧\nhμ : SigmaFinite (μ.trim hm)\nA : μ[fun x ↦ (ENNReal.ofReal (f x)).toReal | m] =ᵐ[μ] μ[f | m]\nB : 0 ≤ᵐ[μ] μ[f | m]\ng : 𝓧 → ℝ≥0∞ := fun x ↦ ENNReal.ofReal (f x)\nI : ∫⁻ (x : 𝓧), g x ∂μ ≠ ∞\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 140,
"column": 32
} | {
"line": 140,
"column": 34
} | {
"line": 141,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 146,
"column": 69
} | {
"line": 146,
"column": 71
} | {
"line": 146,
"column": 71
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 149,
"column": 62
} | {
"line": 149,
"column": 64
} | {
"line": 149,
"column": 64
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 159,
"column": 41
} | {
"line": 159,
"column": 43
} | {
"line": 159,
"column": 44
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.InformationTheory.KullbackLeibler.DataProcessing | {
"line": 120,
"column": 53
} | {
"line": 120,
"column": 55
} | {
"line": 121,
"column": 2
} | [
{
"pp": "𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nf : ℝ → ℝ\nhm : m ≤ m𝓧\nhμν : μ ≪ ν\nhf : StronglyMeasurable f\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_cont_at : ContinuousWithinAt f (Ici 0) 0\nh_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toRe... | [
"𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nf : ℝ → ℝ\nhm : m ≤ m𝓧\nhμν : μ ≪ ν\nhf : StronglyMeasurable f\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_cont_at : ContinuousWithinAt f (Ici 0) 0\nh_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν\nh : I... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.InformationTheory.KullbackLeibler.DataProcessing | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 8
} | {
"line": 121,
"column": 8
} | [
{
"pp": "𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nf : ℝ → ℝ\nhm : m ≤ m𝓧\nhμν : μ ≪ ν\nhf : StronglyMeasurable f\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_cont_at : ContinuousWithinAt f (Ici 0) 0\nh_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toRe... | [
"𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nf : ℝ → ℝ\nhm : m ≤ m𝓧\nhμν : μ ≪ ν\nhf : StronglyMeasurable f\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_cont_at : ContinuousWithinAt f (Ici 0) 0\nh_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν\nh : I... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 181,
"column": 37
} | {
"line": 181,
"column": 39
} | {
"line": 181,
"column": 40
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.InformationTheory.KullbackLeibler.DataProcessing | {
"line": 143,
"column": 52
} | {
"line": 143,
"column": 54
} | {
"line": 143,
"column": 55
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : 𝓧 → 𝓨\nhμν : μ ≪ ν\nhg : Measurable g\na : 𝓧\n⊢ ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal = ν[fun a ↦ ((∂μ/∂ν) a).toReal | Measurabl... | [
"𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : 𝓧 → 𝓨\nhμν : μ ≪ ν\nhg : Measurable g\na : 𝓧\nha : ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal = ν[fun a ↦ ((∂μ/∂ν) a).toReal | MeasurableSpace.co... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.InformationTheory.KullbackLeibler.DataProcessing | {
"line": 143,
"column": 68
} | {
"line": 143,
"column": 70
} | {
"line": 143,
"column": 70
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : 𝓧 → 𝓨\nhμν : μ ≪ ν\nhg : Measurable g\na : 𝓧\nha : ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal = ν[fun a ↦ ((∂μ/∂ν) a).toReal | Measur... | [
"𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : 𝓧 → 𝓨\nhμν : μ ≪ ν\nhg : Measurable g\na : 𝓧\nha : ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal = ν[fun a ↦ ((∂μ/∂ν) a).toReal | MeasurableSpace.co... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 198,
"column": 36
} | {
"line": 198,
"column": 38
} | {
"line": 199,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ s φ\nhφ_cont : UpperSemicontinuousOn φ s\nhf : ∀ᵐ (... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ s φ\nhφ_cont : UpperSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.InformationTheory.KullbackLeibler.DataProcessing | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 49
} | {
"line": 167,
"column": 2
} | [
{
"pp": "case pos\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : 𝓧 → 𝓨\nhg : Measurable g\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\n⊢ ENNReal.ofReal (∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).to... | [
"case pos\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : 𝓧 → 𝓨\nhg : Measurable g\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\n⊢ ENNReal.ofReal (∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | Measu... | conv_rhs => rw [← integral_condExp hg.comap_le] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 233,
"column": 36
} | {
"line": 233,
"column": 38
} | {
"line": 234,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ univ φ\nhφ_cont : UpperSemicontinuous φ\nhf_int : Integrable f... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ univ φ\nhφ_cont : UpperSemicontinuous φ\nhf_int : Integrable f μ\nhφ_int :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.InformationTheory.KullbackLeibler.DataProcessing | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 39
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case h.refine_2\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : 𝓧 → 𝓨\nhg : Measurable g\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\nhf : StronglyMeasurable klFun\nhf_cont : ContinuousWit... | [
"case h.refine_2\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : 𝓧 → 𝓨\nhg : Measurable g\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\nhf : StronglyMeasurable klFun\nhf_cont : ContinuousWithinAt klFun ... | refine ae_of_ae_trim hg.comap_le ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 273,
"column": 50
} | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 53
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\np : ℝ\nhp : 1 ≤ p\nhfint : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : 0 < p\nhm : m ≤ mα\nhμm : SigmaFinite (μ.trim hm)\nhf_int : Integrable f μ... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\np : ℝ\nhp : 1 ≤ p\nhfint : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : 0 < p\nhm : m ≤ mα\nhμm : SigmaFinite (μ.trim hm)\nhf_int : Integrable f μ\nhl : Lower... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 291,
"column": 36
} | {
"line": 291,
"column": 38
} | {
"line": 292,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : FiniteDimensional ℝ E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ univ φ\nhf_int : Integrable f ... | [
"E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : FiniteDimensional ℝ E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ univ φ\nhf_int : Integrable f μ\nhφ_int : ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 154,
"column": 37
} | {
"line": 154,
"column": 58
} | {
"line": 154,
"column": 59
} | [
{
"pp": "ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : CommRing k\ninst✝² : Module k V\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb b₂ : AffineBasis ι k P\nx : P\nhu : IsUnit (b.toMatrix ⇑b₂).det\n⊢ b₂.coords x ᵥ* b.toMatrix ⇑b₂ ᵥ* (b.toMatrix ⇑... | [
"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : CommRing k\ninst✝² : Module k V\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb b₂ : AffineBasis ι k P\nx : P\nhu : IsUnit (b.toMatrix ⇑b₂).det\n⊢ b₂.coords x ᵥ* (b.toMatrix ⇑b₂ * (b.toMatrix ⇑b₂)⁻¹) = b₂.... | Matrix.vecMul_vecMul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 162,
"column": 2
} | {
"line": 167,
"column": 94
} | {
"line": 168,
"column": 2
} | [
{
"pp": "case inr.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : CommRing k\ninst✝² : AddCommGroup V\ninst✝¹ : AddTorsor V P\ninst✝ : Module k V\ns : AffineSubspace k P\nc : P\nr : k\nhr : IsUnit r\nt : k\nht : r * t = 1\nh✝ : (↑s).Nonempty\nh : Nonempty ↥s\np x : P\nhmem : x ∈ s\nheq : (1 - r) • (c -ᵥ ... | [
"case inr.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : CommRing k\ninst✝² : AddCommGroup V\ninst✝¹ : AddTorsor V P\ninst✝ : Module k V\ns : AffineSubspace k P\nc : P\nr : k\nhr : IsUnit r\nt : k\nht : r * t = 1\nh✝ : (↑s).Nonempty\nh : Nonempty ↥s\np x : P\nhmem : x ∈ s\nheq : r • (x -ᵥ c) +ᵥ c = p\n⊢ ∃ ... | · refine ⟨t • (x -ᵥ h.some.val) +ᵥ h.some.val, ?_, ?_⟩
· refine vadd_mem_of_mem_direction ?_ h.some.prop
exact smul_mem _ _ <| vsub_mem_direction hmem h.some.prop
· rw [vadd_vsub_assoc, smul_add, smul_smul, ht, sub_eq_add_neg, add_smul, one_smul, one_smul,
neg_smul, ← smul_neg, neg_vsub_eq_vsub_re... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 249,
"column": 8
} | {
"line": 249,
"column": 21
} | {
"line": 249,
"column": 21
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nthis : ∀ e ∈ transvections R V, e.symm ∈ transvections R V\nh : e.symm ∈ transvections R V\n⊢ e ∈ transvections R V",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nthis : ∀ e ∈ transvections R V, e.symm ∈ transvections R V\nh : e.symm ∈ transvections R V\n⊢ e.symm.symm ∈ transvections R V"
] | ← symm_symm e | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.TensorProduct | {
"line": 76,
"column": 25
} | {
"line": 76,
"column": 38
} | {
"line": 76,
"column": 38
} | [
{
"pp": "R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\nN₁ : Type uN₁\nN₂ : Type uN₂\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : AddCommGroup M₁\ninst✝¹⁴ : AddCommGroup M₂\ninst✝¹³ : AddCommGroup N₁\ninst✝¹² : AddCommGroup N₂\ninst✝¹¹ : Algebra R A\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : Module A M... | [
"R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\nN₁ : Type uN₁\nN₂ : Type uN₂\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : AddCommGroup M₁\ninst✝¹⁴ : AddCommGroup M₂\ninst✝¹³ : AddCommGroup N₁\ninst✝¹² : AddCommGroup N₂\ninst✝¹¹ : Algebra R A\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : Module A M₁\ninst✝⁸ : ... | BilinMap.tmul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 454,
"column": 8
} | {
"line": 454,
"column": 62
} | {
"line": 455,
"column": 8
} | [
{
"pp": "case hd\nV : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe✝ : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_ne_t... | [
"case hd\nV : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe✝ : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_ne_top : (↑e).fi... | have := (LinearMap.ker f).finrank_quotient_add_finrank | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 188,
"column": 75
} | {
"line": 193,
"column": 100
} | {
"line": 195,
"column": 0
} | [
{
"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 : CliffordAlgebra Q\n⊢ (contractLeft d) ((contractLeft d) x) = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cliffor... | [] | by
induction x using CliffordAlgebra.left_induction with
| algebraMap => simp_rw [contractLeft_algebraMap, map_zero]
| add _ _ hx hy => rw [map_add, map_add, hx, hy, add_zero]
| ι_mul _ _ hx =>
rw [contractLeft_ι_mul, map_sub, contractLeft_ι_mul, hx, map_smul, mul_zero, sub_zero, sub_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 588,
"column": 4
} | {
"line": 588,
"column": 67
} | {
"line": 589,
"column": 4
} | [
{
"pp": "case neg\nK : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : Dual K V\nv : V\nhfv : ¬f v = 0\n⊢ LinearMap.det (transvection f v) = 1 + f v",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"LinearMap.t... | [
"case neg\nK : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : Dual K V\nv : V\nhfv : ¬f v = 0\nι : Set V\nb : Basis (↑ι) K V\ni : ↑ι\nhv : v = b i\nhf : f = f v • b.coord i\n⊢ LinearMap.det (transvection f v) = 1 + f v"
] | obtain ⟨ι, b, i, hv, hf⟩ := exists_basis_of_pairing_ne_zero hfv | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 206,
"column": 17
} | {
"line": 206,
"column": 36
} | {
"line": 206,
"column": 37
} | [
{
"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\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (contractLeft d) ((contractLeft d') x✝) = -(contractLeft d') ((contractLeft d) x✝)\n⊢ d' m✝ • (contractLeft d) x✝ -... | [
"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\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (contractLeft d) ((contractLeft d') x✝) = -(contractLeft d') ((contractLeft d) x✝)\n⊢ d' m✝ • (contractLeft d) x✝ + (ι Q) m✝ * ... | sub_sub_eq_add_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 604,
"column": 8
} | {
"line": 604,
"column": 11
} | {
"line": 604,
"column": 12
} | [
{
"pp": "case neg\nK : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : Dual K V\nv : V\nhfv : ¬f v = 0\nι : Set V\nb : Basis (↑ι) K V\ni : ↑ι\nhv : v = b i\nhf : f = f v • b.coord i\nthis : Fintype ↑ι\nx y : ↑ι\n⊢ (Finsupp.single y 1) x ... | [
"case neg\nK : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : Dual K V\nv : V\nhfv : ¬f v = 0\nι : Set V\nb : Basis (↑ι) K V\ni : ↑ι\nhv : v = b i\nhf : f = f v • b.coord i\nthis : Fintype ↑ι\nx y : ↑ι\n⊢ (Finsupp.single y 1) x + f (b y) * ... | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 60
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : 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 : ι) → Mo... | [] | rw [gradedComm_of_tmul_of, zero_mul, uzpow_zero, one_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 197,
"column": 22
} | {
"line": 197,
"column": 43
} | {
"line": 197,
"column": 43
} | [
{
"pp": "R : 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)\ninst✝⁴ : (i : ι) →... | [
"R : 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)\ninst✝⁴ : (i : ι) → Module R (ℬ... | gradedComm_of_tmul_of | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 77,
"column": 40
} | {
"line": 77,
"column": 42
} | {
"line": 77,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\nx✝ : (CliffordAlgebra Q)ˣ\ninst✝ : Invertible 2\nx : (CliffordAlgebra Q)ˣ\nm a : M\nha : (ι Q) a = ↑x\nthis : Invertible ↑x := x.invertible\n⊢ Invertible ((ι Q) a)",
"ppTerm": "?m.1... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\nx✝ : (CliffordAlgebra Q)ˣ\ninst✝ : Invertible 2\nx : (CliffordAlgebra Q)ˣ\nm a : M\nha : (ι Q) a = ↑x\nthis : Invertible ↑x := x.invertible\n⊢ Invertible ↑x"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 83,
"column": 40
} | {
"line": 83,
"column": 42
} | {
"line": 83,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\nx✝ : (CliffordAlgebra Q)ˣ\ninst✝ : Invertible 2\nx : (CliffordAlgebra Q)ˣ\nm a : M\nha : (ι Q) a = ↑x\nthis : Invertible ↑x := x.invertible\n⊢ Invertible ((ι Q) a)",
"ppTerm": "?m.1... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\nx✝ : (CliffordAlgebra Q)ˣ\ninst✝ : Invertible 2\nx : (CliffordAlgebra Q)ˣ\nm a : M\nha : (ι Q) a = ↑x\nthis : Invertible ↑x := x.invertible\n⊢ Invertible ↑x"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 104,
"column": 40
} | {
"line": 104,
"column": 42
} | {
"line": 104,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx✝ x : (CliffordAlgebra Q)ˣ\nb a : M\nha : (ι Q) a = ↑x\nthis : Invertible ↑x := x.invertible\n⊢ Invertible ((ι Q) a)",
"ppTerm": "?m.116",
"assigned": tru... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx✝ x : (CliffordAlgebra Q)ˣ\nb a : M\nha : (ι Q) a = ↑x\nthis : Invertible ↑x := x.invertible\n⊢ Invertible ↑x"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 111,
"column": 40
} | {
"line": 111,
"column": 42
} | {
"line": 111,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx✝ x : (CliffordAlgebra Q)ˣ\nb a : M\nha : (ι Q) a = ↑x\nthis : Invertible ↑x := x.invertible\n⊢ Invertible ((ι Q) a)",
"ppTerm": "?m.166",
"assigned": tru... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx✝ x : (CliffordAlgebra Q)ˣ\nb a : M\nha : (ι Q) a = ↑x\nthis : Invertible ↑x := x.invertible\n⊢ Invertible ↑x"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 83,
"column": 8
} | {
"line": 93,
"column": 77
} | {
"line": 95,
"column": 0
} | [
{
"pp": "case mem.mem_mul.e_a.e_a.mem\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 ... | [] | induction hm₂ using Submodule.pow_induction_on_left' with
| algebraMap =>
rw [AlgHom.commutes, Nat.cast_zero, uzpow_zero, one_smul, Algebra.commutes]
| add _ _ _ _ _ ihx ihy =>
rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]
| mem_mul m₂ hm₂ i x₂ _hx₂ ih₂ =>
obta... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.Goursat | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 14
} | {
"line": 125,
"column": 4
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\nM' : Submodule R M := map (LinearMap.fst R M N) L\nN' : Submodule R N := map (LinearMap.snd R M N) L\nP : ↥L →ₗ[R] ↥M' :... | [
"case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\nM' : Submodule R M := map (LinearMap.fst R M N) L\nN' : Submodule R N := map (LinearMap.snd R M N) L\nP : ↥L →ₗ[R] ↥M' := (LinearMap... | ext ⟨m, n⟩ | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.LinearAlgebra.Matrix.Determinant.Misc | {
"line": 40,
"column": 29
} | {
"line": 40,
"column": 46
} | {
"line": 41,
"column": 8
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin (n + 1)) (Fin n) R\nhv : ∑ j, M j = 0\nj₁ j₂ : Fin (n + 1)\ni : Fin n\nh_ind : (M.submatrix i.castSucc.succAbove id).det = (↑↑i.castSucc).negOnePow • (M.submatrix (Fin.succAbove 0) id).det\n⊢ -(M.submatrix i.succ.succAbove id).det = (↑... | [
"case succ\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin (n + 1)) (Fin n) R\nhv : ∑ j, M j = 0\nj₁ j₂ : Fin (n + 1)\ni : Fin n\nh_ind : (M.submatrix i.castSucc.succAbove id).det = (↑↑i.castSucc).negOnePow • (M.submatrix (Fin.succAbove 0) id).det\n⊢ -1 • (M.submatrix i.succ.succAbove id).det = (↑↑i).negO... | ← neg_one_smul R, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Determinant.Misc | {
"line": 67,
"column": 46
} | {
"line": 67,
"column": 49
} | {
"line": 67,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin n) (Fin (n + 1)) R\nhv : ∀ (i : Fin n), ∑ j, M i j = 0\nj₁ j₂ : Fin (n + 1)\nx✝ : Fin n\n⊢ ∑ x, M x✝ x = 0 x✝",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Com... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin n) (Fin (n + 1)) R\nhv : ∀ (i : Fin n), ∑ j, M i j = 0\nj₁ j₂ : Fin (n + 1)\nx✝ : Fin n\n⊢ 0 = 0 x✝"
] | hv, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 67
} | {
"line": 194,
"column": 4
} | [
{
"pp": "case refine_2\nι : Type u_1\ninst✝ : Finite ι\nN : Submodule ℤ (ι → ℤ)\nn : ℕ\nthis : Fintype ι\nbN : Module.Basis (Fin n) ℤ ↥N\nx✝ : Nonempty (↥N ≃ₗ[ℤ] ι → ℤ)\ne : ↥N ≃ₗ[ℤ] ι → ℤ\n⊢ n = Fintype.card ι",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup... | [
"case refine_2\nι : Type u_1\ninst✝ : Finite ι\nN : Submodule ℤ (ι → ℤ)\nn : ℕ\nthis : Fintype ι\nbN : Module.Basis (Fin n) ℤ ↥N\nx✝ : Nonempty (↥N ≃ₗ[ℤ] ι → ℤ)\ne : ↥N ≃ₗ[ℤ] ι → ℤ\nhc : Fintype.card (Fin n) = Fintype.card ι\n⊢ n = Fintype.card ι"
] | have hc := card_eq_of_linearEquiv ℤ <| bN.equivFun.symm.trans e | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 155,
"column": 10
} | {
"line": 155,
"column": 56
} | {
"line": 156,
"column": 8
} | [
{
"pp": "m : Type u_1\nm' : Type u_2\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix m n R\nB : Matrix m' n R\nhA : A.IsTotallyUnimodular\nhB : ∀ (i : m'), ∃ j s, B i = Pi.single j ↑s\nk : ℕ\nih :\n ∀ (f : Fin k → m ⊕ m') (g : Fin k → n),\n Function.Injective f → Function... | [] | · simp [Pi.single_eq_of_ne, hg.ne_iff.mpr hxy] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Card | {
"line": 50,
"column": 6
} | {
"line": 58,
"column": 84
} | {
"line": 60,
"column": 0
} | [
{
"pp": "case succ\nK : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Finite V\nk : ℕ\nih : k ≤ n → card { s // LinearIndependent K s } = ∏ i, (q ^ n - q ^ ↑i)\nhk : k + 1 ≤ n\n⊢ card { s // LinearIndependent K s } = ∏ i, (q ^ n - q ^ ... | [] | have (s : { s : Fin k → V // LinearIndependent K s }) :
card ((Submodule.span K (Set.range (s : Fin k → V)))ᶜ : Set (V)) =
(q) ^ n - (q) ^ k := by
rw [card_compl_set, Module.card_eq_pow_finrank (K := K)
(V := ((Submodule.span K (Set.range (s : Fin k → V))) : Set (V)))]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Card | {
"line": 50,
"column": 6
} | {
"line": 58,
"column": 84
} | {
"line": 60,
"column": 0
} | [
{
"pp": "case succ\nK : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Finite V\nk : ℕ\nih : k ≤ n → card { s // LinearIndependent K s } = ∏ i, (q ^ n - q ^ ↑i)\nhk : k + 1 ≤ n\n⊢ card { s // LinearIndependent K s } = ∏ i, (q ^ n - q ^ ... | [] | have (s : { s : Fin k → V // LinearIndependent K s }) :
card ((Submodule.span K (Set.range (s : Fin k → V)))ᶜ : Set (V)) =
(q) ^ n - (q) ^ k := by
rw [card_compl_set, Module.card_eq_pow_finrank (K := K)
(V := ((Submodule.span K (Set.range (s : Fin k → V))) : Set (V)))]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 43
} | {
"line": 93,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\n⊢ ∀ (c : ℕ), (-A).den ∣ c ↔ A.den ∣ c",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"congrArg",
"Matrix",
"Rat",
"semigroupDvd",
"Rat.den",
"... | [] | simp [den_dvd_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 43
} | {
"line": 93,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\n⊢ ∀ (c : ℕ), (-A).den ∣ c ↔ A.den ∣ c",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"congrArg",
"Matrix",
"Rat",
"semigroupDvd",
"Rat.den",
"... | [] | simp [den_dvd_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 43
} | {
"line": 93,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\n⊢ ∀ (c : ℕ), (-A).den ∣ c ↔ A.den ∣ c",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"congrArg",
"Matrix",
"Rat",
"semigroupDvd",
"Rat.den",
"... | [] | simp [den_dvd_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Permanent | {
"line": 38,
"column": 17
} | {
"line": 38,
"column": 67
} | {
"line": 39,
"column": 2
} | [
{
"pp": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nd : n → R\nσ : Perm n\nx✝ : σ ∈ univ\nhσ : σ ≠ 1\nx : n\nhx : ¬σ x = 1 x\n⊢ ∏ i, diagonal d (σ i) i = 0",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"E... | [] | exact Finset.prod_eq_zero (mem_univ x) (if_neg hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Permanent | {
"line": 38,
"column": 17
} | {
"line": 38,
"column": 67
} | {
"line": 39,
"column": 2
} | [
{
"pp": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nd : n → R\nσ : Perm n\nx✝ : σ ∈ univ\nhσ : σ ≠ 1\nx : n\nhx : ¬σ x = 1 x\n⊢ ∏ i, diagonal d (σ i) i = 0",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"E... | [] | exact Finset.prod_eq_zero (mem_univ x) (if_neg hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Permanent | {
"line": 38,
"column": 17
} | {
"line": 38,
"column": 67
} | {
"line": 39,
"column": 2
} | [
{
"pp": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nd : n → R\nσ : Perm n\nx✝ : σ ∈ univ\nhσ : σ ≠ 1\nx : n\nhx : ¬σ x = 1 x\n⊢ ∏ i, diagonal d (σ i) i = 0",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"E... | [] | exact Finset.prod_eq_zero (mem_univ x) (if_neg hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Permanent | {
"line": 111,
"column": 52
} | {
"line": 111,
"column": 77
} | {
"line": 112,
"column": 4
} | [
{
"pp": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nM : Matrix n n R\nj : n\nc : R\nu : n → R\n⊢ (Mᵀ.updateCol j (c • u)).permanent = c * (M.updateRow j u).permanent",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nM : Matrix n n R\nj : n\nc : R\nu : n → R\n⊢ c * (Mᵀ.updateCol j u).permanent = c * (M.updateRow j u).permanent"
] | permanent_updateCol_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PiTensorProduct | {
"line": 73,
"column": 52
} | {
"line": 73,
"column": 54
} | {
"line": 73,
"column": 54
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalNonAssocSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\na₁ a₂ a₃ : (i : ι) → A i\nha : SemiconjBy a₁ a₂ ... | [
"ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalNonAssocSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\na₁ a₂ a₃ : (i : ι) → A i\nha : SemiconjBy a₁ a₂ a₃\n⊢ (PiTen... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PiTensorProduct | {
"line": 297,
"column": 56
} | {
"line": 298,
"column": 30
} | {
"line": 300,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : Fintype ι\nx : ι → R\n⊢ (constantBaseRingEquiv ι R) ((tprod R) x) = ∏ i, x i",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"PiTensorProduct.instModule",
"PiTensorProduct.lift.tprod",
"Semiring.toModul... | [] | by
simp [constantBaseRingEquiv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 238,
"column": 27
} | {
"line": 239,
"column": 28
} | {
"line": 240,
"column": 4
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Submodule K V\nx : V\nthis : x = 0 → x ∈ s\n⊢ x ∈\n (fun s ↦\n { carrier := {x | ∀ (h : x ≠ 0), Projectivization.mk K x h ∈ s.carrier}, add_mem' := ⋯, zero_mem' := ⋯,\n smul_me... | [] | by
simpa [imp_iff_not_or] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.QuadraticForm.Basis | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 32
} | {
"line": 73,
"column": 33
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : DecidableEq ι\nQ : QuadraticMap R M N\nbm : Basis ι R M\nx : M\n⊢ (linearCombination R (⇑Q ∘ ⇑bm)) (bm.repr x * bm.repr x) +\n... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : DecidableEq ι\nQ : QuadraticMap R M N\nbm : Basis ι R M\nx : M\n⊢ Q ((linearCombination R ⇑bm) (bm.repr x)) = Q x"
] | ← apply_linearCombination, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 443,
"column": 26
} | {
"line": 443,
"column": 73
} | {
"line": 443,
"column": 73
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Module.Free R V\ninst✝ : Module.Finite R V\ng : ↥(Subgroup.center (SpecialLinearGroup R V))\nhR : Nontrivial R\nhV : 1 ≤ Module.finrank R V\nr : R := ⋯.choose\nhr : r ^ max (Module.finrank R V) 1 = 1... | [] | by simp [mem_rootsOfUnity, ← Units.val_inj, hr] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 68
} | {
"line": 121,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\nP : RootPairing ι R M N\ninst✝⁶ : P.IsCrystallographic\ninst✝⁵ : P.IsReduced\ninst✝⁴ : P.IsIrreducible\ninst✝³ : Finite ι\ninst✝² : ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\nP : RootPairing ι R M N\ninst✝⁶ : P.IsCrystallographic\ninst✝⁵ : P.IsReduced\ninst✝⁴ : P.IsIrreducible\ninst✝³ : Finite ι\ninst✝² : CharZero R\n... | have aux₁ := P.forall_pairingIn_eq_swap_or.resolve_left this i j | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 195,
"column": 58
} | {
"line": 195,
"column": 90
} | {
"line": 196,
"column": 2
} | [
{
"pp": "ι : 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 : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ni j : ι\ninst✝ : P.IsNotG2\nh : P.root i - P.roo... | [] | simpa [← sub_eq_add_neg] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 218,
"column": 4
} | {
"line": 226,
"column": 21
} | {
"line": 227,
"column": 4
} | [
{
"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... | [
"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.IsCrystallographic\nb : ... | obtain ⟨l, hl⟩ : ∃ l, χ (h' l) ≠ 0 := by
replace hw₀ : genWeightSpace (b.support ⊕ ι → K) χ ≠ ⊥ := by
contrapose hw₀; rw [LieSubmodule.eq_bot_iff] at hw₀; exact hw₀ _ hw
let χ' : H →ₗ[K] K := (Weight.mk χ hw₀).toLinear
replace hχ : χ' ≠ 0 := by contrapose hχ; ext x; simpa using! LinearMap.cong... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 172,
"column": 8
} | {
"line": 173,
"column": 91
} | {
"line": 174,
"column": 8
} | [
{
"pp": "ι : 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 : P.Base\ninst... | [
"ι : 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 : P.Base\ninst✝¹ : Fintype... | replace contra : P.root x = -(P.root i + P.root i) := by
simpa [neg_eq_iff_add_eq_zero, ← add_assoc, add_eq_zero_iff_eq_neg'] using contra | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.LinearAlgebra.RootSystem.OfBilinear | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 32
} | {
"line": 67,
"column": 4
} | [
{
"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\nr : R\na : M\n⊢ (fun x_1 ↦ (B x) x * x_1) (Exists.choose ⋯) = (fun x_1 ↦ (B x) x * x_1) ((RingHom.id R) r • Exists.choose ⋯)",
"ppTerm": "?m.153",
"... | [
"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\nr : R\na : M\n⊢ (B x) x * Exists.choose ⋯ = (B x) x * r • Exists.choose ⋯"
] | simp only [RingHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.OfBilinear | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 38
} | {
"line": 91,
"column": 39
} | [
{
"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\nhSB : B.IsSymm\ny z : M\n⊢ (B x) x * (B y) z - 2 * (B x) y * (B x) z =\n (B x) x * (B y) z + (2 * (B x) z * (B y) x - 2 * (B x) z * (coroot B hx) y * (B ... | [
"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\nhSB : B.IsSymm\ny z : M\n⊢ (B x) x * (B y) z - 2 * (B x) y * (B x) z =\n (B x) x * (B y) z + (2 * (B x) z * (RingHom.id R) ((B x) y) - 2 * (B x) z * (coroot B hx) y ... | ← hSB.eq x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.OfBilinear | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 19
} | {
"line": 181,
"column": 20
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsReflexive R M\nB : M →ₗ[R] M →ₗ[R] R\nhNB : B.Nondegenerate\nhSB : B.IsSymm\nh2 : IsRegular 2\nx y : ↑{x | B.IsReflective x}\nz : M\n⊢ 2 * (B ↑y) ↑x * (2 * (B ↑x) z) = 2 * (2 * (B ↑x) ↑y) * (B z) ↑x... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsReflexive R M\nB : M →ₗ[R] M →ₗ[R] R\nhNB : B.Nondegenerate\nhSB : B.IsSymm\nh2 : IsRegular 2\nx y : ↑{x | B.IsReflective x}\nz : M\n⊢ 2 * (RingHom.id R) ((B ↑x) ↑y) * (2 * (B ↑x) z) = 2 * (2 * (B ↑x) ↑y) * (B ... | ← hSB.eq x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SesquilinearForm.Star | {
"line": 33,
"column": 4
} | {
"line": 34,
"column": 52
} | {
"line": 35,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : StarRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : M →ₗ⋆[R] M →ₗ[R] R\nι : Type u_4\nb : Basis ι R M\nh : ∀ (i j : ι), star ((B (b i)) (b j)) = (B (b j)) (b i)\nx y : M\nfx : M → R\ntx : Finset M\nix : ↑tx ⊆ Set.range ⇑b\nhx : ∑ a... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : StarRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : M →ₗ⋆[R] M →ₗ[R] R\nι : Type u_4\nb : Basis ι R M\nh : ∀ (i j : ι), star ((B (b i)) (b j)) = (B (b j)) (b i)\nx y : M\nfx : M → R\ntx : Finset M\nix : ↑tx ⊆ Set.range ⇑b\nhx : ∑ a ∈ tx, fx a ... | obtain ⟨fy, ty, iy, -, hy⟩ := Submodule.mem_span_iff_exists_finset_subset.1
(by simp : y ∈ Submodule.span R (Set.range b)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 385,
"column": 4
} | {
"line": 386,
"column": 92
} | {
"line": 387,
"column": 4
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrystallographic\n... | simp_rw [← hc, lie_sum, lie_smul, aux, smul_comm (M := K), ← smul_assoc, ← Finset.sum_smul,
zsmul_eq_mul, mul_comm, ← LieSubmodule.mem_toSubmodule, U.smul_mem_iff contra] at this | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.