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.Distributions.Fernique | {
"line": 590,
"column": 2
} | {
"line": 591,
"column": 51
} | {
"line": 592,
"column": 2
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : SecondCountableTopology E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod... | [
"case neg.refine_1\nE : Type u_1\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : SecondCountableTopology E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ\... | refine integrable_of_le_of_le (g₁ := 0) (g₂ := fun _ ↦ rexp (b ^ 2)) (by fun_prop)
?_ ?_ (integrable_const _) (integrable_const _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 56
} | {
"line": 97,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDF f κ ν\na✝ : α\nq : ℚ\na : β\n⊢ IsRatStieltjesPoint f (a✝, a) → ↑(stieltjesOfMeasurableRat f ⋯ (a✝, a)) ↑q = f (a✝, a) q",
"ppTerm": "?m.52",
... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDF f κ ν\na✝ : α\nq : ℚ\na : β\nha : IsRatStieltjesPoint f (a✝, a)\n⊢ ↑(stieltjesOfMeasurableRat f ⋯ (a✝, a)) ↑q = f (a✝, a) q"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 73
} | {
"line": 331,
"column": 2
} | [
{
"pp": "α : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\nx : ℝ\n⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Ioi",
... | [
"α : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\nx : ℝ\n⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x ↔\n Tendsto (stieltjesFunctionAux f a) (𝓝[>] x) (𝓝 (sInf (stieltjesFunctionAux f a '' Ioi x)))"
] | convert! Monotone.tendsto_nhdsGT (monotone_stieltjesFunctionAux hf a) x | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 75
} | {
"line": 401,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\nh_exists : ∀ (x : ℝ), ∃ q, x - 1 < ↑q ∧ ↑q < x\nqs : ℝ → ℚ := fun x ↦ ⋯.choose\nhqs_tendsto : Tendsto qs atTop atTop\nx : ℝ\n⊢ ↑(hf.stieltjesFunction a) ↑(qs x) ≤ ↑(hf.stieltjesFunction a) x",
"ppTerm": "?m.16... | [] | exact (hf.stieltjesFunction a).mono (le_of_lt (h_exists x).choose_spec.2) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 74
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ninst✝ : IsFiniteMeasure ρ\n⊢ ∫⁻ (x : α), preCDF ρ r x ∂ρ.fst = (ρ.IicSnd ↑r) univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.IicSnd",
"ProbabilityTheory.preCDF",
"Eq.mpr... | [] | rw [← setLIntegral_univ, setLIntegral_preCDF_fst ρ r MeasurableSet.univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 74
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ninst✝ : IsFiniteMeasure ρ\n⊢ ∫⁻ (x : α), preCDF ρ r x ∂ρ.fst = (ρ.IicSnd ↑r) univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.IicSnd",
"ProbabilityTheory.preCDF",
"Eq.mpr... | [] | rw [← setLIntegral_univ, setLIntegral_preCDF_fst ρ r MeasurableSet.univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 74
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ninst✝ : IsFiniteMeasure ρ\n⊢ ∫⁻ (x : α), preCDF ρ r x ∂ρ.fst = (ρ.IicSnd ↑r) univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.IicSnd",
"ProbabilityTheory.preCDF",
"Eq.mpr... | [] | rw [← setLIntegral_univ, setLIntegral_preCDF_fst ρ r MeasurableSet.univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 186,
"column": 44
} | {
"line": 186,
"column": 46
} | {
"line": 187,
"column": 4
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ns : Set α\nhs : MeasurableSet s\ninst✝ : IsFiniteMeasure ρ\na : α\n⊢ (∀ (r : ℚ), preCDF ρ r a ≤ 1) → preCDF ρ r a < ∞",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.preCDF",
"Rat",
... | [
"α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ns : Set α\nhs : MeasurableSet s\ninst✝ : IsFiniteMeasure ρ\na : α\nha : ∀ (r : ℚ), preCDF ρ r a ≤ 1\n⊢ preCDF ρ r a < ∞"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 200,
"column": 44
} | {
"line": 200,
"column": 46
} | {
"line": 200,
"column": 47
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\nx : ℚ\nt : Set α\nx✝¹ : MeasurableSet t\nx✝ : ρ.fst t ≠ ∞\na : α\n⊢ (∀ (r : ℚ), preCDF ρ r a ≤ 1) → ENNReal.ofReal (preCDF ρ x a).toReal ≤ 1",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\nx : ℚ\nt : Set α\nx✝¹ : MeasurableSet t\nx✝ : ρ.fst t ≠ ∞\na : α\nha : ∀ (r : ℚ), preCDF ρ r a ≤ 1\n⊢ ENNReal.ofReal (preCDF ρ x a).toReal ≤ 1"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 212,
"column": 44
} | {
"line": 212,
"column": 46
} | {
"line": 213,
"column": 4
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\na✝ : Unit\nq : ℚ\na : α\n⊢ (∀ (r : ℚ), preCDF ρ r a ≤ 1) → (preCDF ρ q a).toReal ≤ 1",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.preCDF",
"Rat",
"LE.le",... | [
"α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\na✝ : Unit\nq : ℚ\na : α\nha : ∀ (r : ℚ), preCDF ρ r a ≤ 1\n⊢ (preCDF ρ q a).toReal ≤ 1"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 278,
"column": 61
} | {
"line": 278,
"column": 63
} | {
"line": 278,
"column": 64
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\nr : ℚ\na : α\n⊢ ↑(condCDF ρ a) ↑r = (preCDF ρ r a).toReal →\n (∀ (r : ℚ), preCDF ρ r a ≤ 1) → ENNReal.ofReal (↑(condCDF ρ a) ↑r) = preCDF ρ r a",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\nr : ℚ\na : α\nha : ↑(condCDF ρ a) ↑r = (preCDF ρ r a).toReal\n⊢ (∀ (r : ℚ), preCDF ρ r a ≤ 1) → ENNReal.ofReal (↑(condCDF ρ a) ↑r) = preCDF ρ r a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 64
} | {
"line": 341,
"column": 2
} | [
{
"pp": "case hsm\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\ns : Set α\nhs : MeasurableSet s\nt : ℚ\n⊢ ∀ (i : { r' // t < r' }), NullMeasurableSet (s ×ˢ Iic ↑↑i) ρ",
"ppTerm": "?hsm",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"instC... | [] | exact fun _ => (hs.prod measurableSet_Iic).nullMeasurableSet | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 64
} | {
"line": 341,
"column": 2
} | [
{
"pp": "case hsm\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\ns : Set α\nhs : MeasurableSet s\nt : ℚ\n⊢ ∀ (i : { r' // t < r' }), NullMeasurableSet (s ×ˢ Iic ↑↑i) ρ",
"ppTerm": "?hsm",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"instC... | [] | exact fun _ => (hs.prod measurableSet_Iic).nullMeasurableSet | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 64
} | {
"line": 341,
"column": 2
} | [
{
"pp": "case hsm\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\ns : Set α\nhs : MeasurableSet s\nt : ℚ\n⊢ ∀ (i : { r' // t < r' }), NullMeasurableSet (s ×ˢ Iic ↑↑i) ρ",
"ppTerm": "?hsm",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"instC... | [] | exact fun _ => (hs.prod measurableSet_Iic).nullMeasurableSet | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 347,
"column": 2
} | {
"line": 359,
"column": 69
} | {
"line": 360,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\nq : ℚ\nA : Set β\nhA : MeasurableSet A\n⊢ ∫ (t : β) in A, ⨅ r... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\nq : ℚ\nA : Set β\nhA : MeasurableSet A\n⊢ (κ a).real (A ×ˢ Iic ↑q) ≤ ∫ (t... | · have h : ∀ r : Ioi q, ∫ t in A, ⨅ r' : Ioi q, f (a, t) r' ∂(ν a)
≤ (κ a).real (A ×ˢ Iic (r : ℝ)) := by
intro r
rw [← hf.setIntegral a hA]
refine setIntegral_mono_ae ?_ ?_ ?_
· exact (hf.integrable_iInf_rat_gt _ _).integrableOn
· exact (hf.integrable _ _).integrableOn
· filt... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 493,
"column": 33
} | {
"line": 493,
"column": 58
} | {
"line": 493,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set β\nhs : MeasurableSet s\nh_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Iic ↑q\nh_dir_prod ... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set β\nhs : MeasurableSet s\nh_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Iic ↑q\nh_dir_prod : Directed (... | h_dir_prod.measure_iUnion | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.Composition.Lemmas | {
"line": 105,
"column": 23
} | {
"line": 105,
"column": 56
} | {
"line": 105,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(κ ∥ₖ Kernel.id... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(κ ∥ₖ Kernel.id) ∘ₘ map Pro... | Measure.deterministic_comp_eq_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.Lemmas | {
"line": 109,
"column": 23
} | {
"line": 109,
"column": 56
} | {
"line": 109,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(Kernel.determi... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ map Prod.swap (⇑(Kernel.id ∥... | Measure.deterministic_comp_eq_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.StandardBorel | {
"line": 278,
"column": 31
} | {
"line": 278,
"column": 33
} | {
"line": 279,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nκ : Kernel α (β × Ω)\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) ℝ\ninst✝ : IsSFiniteKernel η\nhη : (κ.map (Prod.map id (embeddingRea... | [
"α : Type u_1\nβ : Type u_2\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nκ : Kernel α (β × Ω)\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) ℝ\ninst✝ : IsSFiniteKernel η\nhη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 35
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass β\nhY : Measurable Y\nx : β\nhX : (... | [] | · rwa [Measure.fst_map_prodMk hY] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 95,
"column": 33
} | {
"line": 95,
"column": 35
} | {
"line": 95,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nX' : α → β\nY' : α → Ω\nhY : Y =ᵐ[μ] Y'\nhX : X =ᵐ[μ] X'\na : α\n⊢ X ... | [
"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nX' : α → β\nY' : α → Ω\nhY : Y =ᵐ[μ] Y'\nhX : X =ᵐ[μ] X'\na : α\nha : X a = X' a\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 515,
"column": 4
} | {
"line": 515,
"column": 57
} | {
"line": 516,
"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 : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nA : Set γ\nhA : MeasurableSet ... | [] | exact this ▸ setIntegral_densityProcess hκν _ _ hs hA | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 515,
"column": 4
} | {
"line": 515,
"column": 57
} | {
"line": 516,
"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 : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nA : Set γ\nhA : MeasurableSet ... | [] | exact this ▸ setIntegral_densityProcess hκν _ _ hs hA | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 515,
"column": 4
} | {
"line": 515,
"column": 57
} | {
"line": 516,
"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 : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nA : Set γ\nhA : MeasurableSet ... | [] | exact this ▸ setIntegral_densityProcess hκν _ _ hs hA | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 51
} | {
"line": 38,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : DecidableLE ι\nX : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b c : ι\ni : ↥(Ioc a c)\nh : ↑i ≤ b\n⊢ Measurable fun c_1 ↦ IocProdIoc a b c c_1 i",
"ppTerm": "?pos✝",
"assigned": true,
"used... | [] | simpa [IocProdIoc, h] using measurable_fst.eval | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 51
} | {
"line": 38,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : DecidableLE ι\nX : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b c : ι\ni : ↥(Ioc a c)\nh : ↑i ≤ b\n⊢ Measurable fun c_1 ↦ IocProdIoc a b c c_1 i",
"ppTerm": "?pos✝",
"assigned": true,
"used... | [] | simpa [IocProdIoc, h] using measurable_fst.eval | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 51
} | {
"line": 38,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : DecidableLE ι\nX : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b c : ι\ni : ↥(Ioc a c)\nh : ↑i ≤ b\n⊢ Measurable fun c_1 ↦ IocProdIoc a b c c_1 i",
"ppTerm": "?pos✝",
"assigned": true,
"used... | [] | simpa [IocProdIoc, h] using measurable_fst.eval | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Integral | {
"line": 44,
"column": 28
} | {
"line": 44,
"column": 30
} | {
"line": 45,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : α → β → E\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, f a =ᵐ[κ a] g a\na✝ : α\n⊢ f a✝ =ᵐ[κ a✝] g a✝ → ∫ (b : β), f a✝ b ∂κ a✝ = ∫ (b : β), g a✝ ... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : α → β → E\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, f a =ᵐ[κ a] g a\na✝ : α\nha : f a✝ =ᵐ[κ a✝] g a✝\n⊢ ∫ (b : β), f a✝ b ∂κ a✝ = ∫ (b : β), g a✝ b ∂κ a✝"... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Integral | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 46
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nη : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → E\n⊢ ∫ (b : β), g b ∂(piecewise hs κ η)... | [] | simp_rw [piecewise_apply]; split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Integral | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 46
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nη : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → E\n⊢ ∫ (b : β), g b ∂(piecewise hs κ η)... | [] | simp_rw [piecewise_apply]; split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Integral | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 46
} | {
"line": 119,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nη : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → E\nt : Set β\n⊢ ∫ (b : β) in t, g b ∂(p... | [] | simp_rw [piecewise_apply]; split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Integral | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 46
} | {
"line": 119,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nη : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → E\nt : Set β\n⊢ ∫ (b : β) in t, g b ∂(p... | [] | simp_rw [piecewise_apply]; split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 99,
"column": 2
} | {
"line": 100,
"column": 59
} | {
"line": 102,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\nu : Set ι\np : ↑I\nS : Set (Set ι)\n⊢ setBer(u, p) S = setBer(u, p) {s | s ∈ S ∧ s ⊆ u}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Probability.Distributions.SetBernoulli.0.ProbabilityTheory.setBernoulli_apply_eq_apply... | [] | apply (measure_eq_measure_of_null_sdiff (by grind) ?_).symm
exact Measure.mono_null (by grind) setBernoulli_ae_subset | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 99,
"column": 2
} | {
"line": 100,
"column": 59
} | {
"line": 102,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\nu : Set ι\np : ↑I\nS : Set (Set ι)\n⊢ setBer(u, p) S = setBer(u, p) {s | s ∈ S ∧ s ⊆ u}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Probability.Distributions.SetBernoulli.0.ProbabilityTheory.setBernoulli_apply_eq_apply... | [] | apply (measure_eq_measure_of_null_sdiff (by grind) ?_).symm
exact Measure.mono_null (by grind) setBernoulli_ae_subset | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 87
} | {
"line": 140,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\nu : Set ι\nhu : u.Finite\np : ↑I\nk : ℕ\nthis : {s | s ⊆ u ∧ s.ncard ∈ {k}}.Finite\ns : Set ι\nhs : s ∈ this.toFinset\n⊢ setBer(u, p).real {s} = ↑p ^ k * (1 - ↑p) ^ (u.ncard - k)",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
"_private.Ma... | [
"ι : Type u_1\ninst✝ : Countable ι\nu : Set ι\nhu : u.Finite\np : ↑I\nk : ℕ\nthis : {s | s ⊆ u ∧ s.ncard ∈ {k}}.Finite\ns : Set ι\nhs : s ⊆ u ∧ s.ncard = k\n⊢ setBer(u, p).real {s} = ↑p ^ k * (1 - ↑p) ^ (u.ncard - k)"
] | simp only [Set.mem_singleton_iff, Set.Finite.mem_toFinset, Set.mem_ofPred_eq] at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.CondVar | {
"line": 129,
"column": 40
} | {
"line": 132,
"column": 11
} | {
"line": 134,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nm₀ m : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhm : m ≤ m₀\ninst✝ : IsFiniteMeasure μ\nhX : MemLp X 2 μ\n⊢ Var[X; μ | m] ≤ᵐ[μ] μ[X ^ 2 | m]",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"MeasureTheory.ae",
... | [] | by
filter_upwards [condVar_ae_eq_condExp_sq_sub_sq_condExp hm hX] with ω hω
dsimp at hω
nlinarith | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Decision.Risk.Defs | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 43
} | {
"line": 129,
"column": 0
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_3\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nP : Kernel Θ 𝓧\ninst✝ : IsEmpty 𝓧\n⊢ minimaxRisk ℓ P = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"bot_nonempty",
"Measur... | [] | simp [minimaxRisk, Subsingleton.elim P 0] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Decision.Risk.Defs | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 43
} | {
"line": 129,
"column": 0
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_3\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nP : Kernel Θ 𝓧\ninst✝ : IsEmpty 𝓧\n⊢ minimaxRisk ℓ P = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"bot_nonempty",
"Measur... | [] | simp [minimaxRisk, Subsingleton.elim P 0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Decision.Risk.Defs | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 43
} | {
"line": 129,
"column": 0
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_3\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nP : Kernel Θ 𝓧\ninst✝ : IsEmpty 𝓧\n⊢ minimaxRisk ℓ P = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"bot_nonempty",
"Measur... | [] | simp [minimaxRisk, Subsingleton.elim P 0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.WithDensity | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 81
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nf : α → β → ℝ≥0∞\nhf : Measurable (Function.uncurry f)\na : α\ns : Set β\na✝ : MeasurableSet s\n⊢ (((κ + η).withDensity f) a) s = ((κ.withDensit... | [] | simp only [Kernel.withDensity_apply _ hf, add_apply, withDensity_add_measure] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ProductMeasure | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 87
} | {
"line": 161,
"column": 2
} | [
{
"pp": "X : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\ninst✝ : ∀ (n : ℕ), SigmaFinite (μ n)\nn : ℕ\ns : (i : ↥(Ioc n (n + 1))) → Set (X ↑i)\nhs : ∀ (i : ↥(Ioc n (n + 1))), MeasurableSet (s i)\n⊢ (map (⇑(piSingleton n)) (μ (n + 1))) (Set.univ.pi s) = ∏ i, (μ ↑i) (s i)",
... | [
"X : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\ninst✝ : ∀ (n : ℕ), SigmaFinite (μ n)\nn : ℕ\ns : (i : ↥(Ioc n (n + 1))) → Set (X ↑i)\nhs : ∀ (i : ↥(Ioc n (n + 1))), MeasurableSet (s i)\nthis : Subsingleton ↥(Ioc n (n + 1))\n⊢ (map (⇑(piSingleton n)) (μ (n + 1))) (Set.univ.pi s)... | have : Subsingleton (Ioc n (n + 1)) := by rw [Nat.Ioc_succ_singleton]; infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 290,
"column": 2
} | {
"line": 291,
"column": 55
} | {
"line": 292,
"column": 2
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\nx : γ\nhx : κ.rnDerivAux (κ + η) a x < 1\n⊢ ENNReal.ofReal (κ.rnDerivAux (κ + η) a x) -\n ... | [
"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\nx : γ\nhx : κ.rnDerivAux (κ + η) a x < 1\n⊢ 1 - κ.rnDerivAux (κ + η) a x ≠ 0",
"α : Type u_1\nγ : Ty... | rw [← ENNReal.ofReal_div_of_pos, div_eq_inv_mul, ← ENNReal.ofReal_mul, ← mul_assoc,
mul_inv_cancel₀, one_mul, tsub_self, Pi.zero_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 315,
"column": 8
} | {
"line": 315,
"column": 61
} | {
"line": 316,
"column": 8
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\ns : Set γ\nhsm : MeasurableSet s\nhs : s ⊆ κ.mutuallySingularSetSlice η a\nhs' : ∀ x ∈ s, 1 ... | [
"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\ns : Set γ\nhsm : MeasurableSet s\nhs : s ⊆ κ.mutuallySingularSetSlice η a\nhs' : ∀ x ∈ s, 1 ≤ κ.rnDerivA... | filter_upwards [rnDerivAux_le_one h_le] with x hx hxs | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 382,
"column": 2
} | {
"line": 392,
"column": 49
} | {
"line": 394,
"column": 0
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\n⊢ η.withDensity (κ.rnDeriv η) + κ.singularPart η = κ",
"ppTerm": "?m.40",
"assigned": true,... | [] | ext a s hs
rw [← inter_union_sdiff s (mutuallySingularSetSlice κ η a)]
simp only [FunLike.coe_add, Pi.add_apply, Measure.coe_add]
have hm := measurableSet_mutuallySingularSetSlice κ η a
simp only [measure_union (Disjoint.mono inter_subset_right le_rfl disjoint_sdiff_right)
(hs.diff hm)]
rw [singularPart_o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 382,
"column": 2
} | {
"line": 392,
"column": 49
} | {
"line": 394,
"column": 0
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\n⊢ η.withDensity (κ.rnDeriv η) + κ.singularPart η = κ",
"ppTerm": "?m.40",
"assigned": true,... | [] | ext a s hs
rw [← inter_union_sdiff s (mutuallySingularSetSlice κ η a)]
simp only [FunLike.coe_add, Pi.add_apply, Measure.coe_add]
have hm := measurableSet_mutuallySingularSetSlice κ η a
simp only [measure_union (Disjoint.mono inter_subset_right le_rfl disjoint_sdiff_right)
(hs.diff hm)]
rw [singularPart_o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 445,
"column": 6
} | {
"line": 445,
"column": 16
} | {
"line": 445,
"column": 16
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = (κ a) (κ.mutuallySingul... | [
"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = (κ a) (κ.mutuallySingularSetSlice η... | ← h_eq_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 457,
"column": 6
} | {
"line": 457,
"column": 16
} | {
"line": 457,
"column": 16
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ((η.withDensity (κ.rnDeriv η)) a) (κ.mutuallySingularSetSlice η a)ᶜ = (κ a) (κ.mu... | [
"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ((η.withDensity (κ.rnDeriv η)) a) (κ.mutuallySingularSetSlice η a)ᶜ = (κ a) (κ.mutuallySingul... | ← h_eq_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous | {
"line": 56,
"column": 32
} | {
"line": 56,
"column": 34
} | {
"line": 57,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ninst✝² : IsFiniteKernel κ\ninst✝¹ : IsFiniteKernel η\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated α β\nμ ν : Measure α\nhκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a\nhμ : SFinite μ\nhν : SFinite ν\ns : Set (α × β... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ninst✝² : IsFiniteKernel κ\ninst✝¹ : IsFiniteKernel η\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated α β\nμ ν : Measure α\nhκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a\nhμ : SFinite μ\nhν : SFinite ν\ns : Set (α × β) := κ.mutua... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous | {
"line": 74,
"column": 33
} | {
"line": 74,
"column": 35
} | {
"line": 75,
"column": 4
} | [
{
"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✝³ : IsFiniteKernel κ\ninst✝² : IsFiniteKernel η\ninst✝¹ : MeasurableSpace.CountableOrCountablyGenerated α β\ninst✝ : SFinite μ\nh : μ ⊗ₘ κ ≪ ν ⊗ₘ η\nthis : ∀ᵐ (a : α) ∂μ, (κ.singularPart ... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ η : Kernel α β\ninst✝³ : IsFiniteKernel κ\ninst✝² : IsFiniteKernel η\ninst✝¹ : MeasurableSpace.CountableOrCountablyGenerated α β\ninst✝ : SFinite μ\nh : μ ⊗ₘ κ ≪ ν ⊗ₘ η\nthis : ∀ᵐ (a : α) ∂μ, (κ.singularPart η) a = 0\na ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.CompProdEqIff | {
"line": 60,
"column": 33
} | {
"line": 60,
"column": 35
} | {
"line": 60,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ η : Kernel α β\ninst✝³ : MeasurableSpace.CountableOrCountablyGenerated α β\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\nh : μ ⊗ₘ κ = μ ⊗ₘ η\nh_ac : ∀ᵐ (a : α) ∂μ, κ a ≪ η a\... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ η : Kernel α β\ninst✝³ : MeasurableSpace.CountableOrCountablyGenerated α β\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\nh : μ ⊗ₘ κ = μ ⊗ₘ η\nh_ac : ∀ᵐ (a : α) ∂μ, κ a ≪ η a\na : α\nha :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.ProductMeasure | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 87
} | {
"line": 431,
"column": 2
} | [
{
"pp": "ι✝ : Type u_1\nX✝ : ι✝ → Type u_2\nmX✝ : (i : ι✝) → MeasurableSpace (X✝ i)\nμ✝ : (i : ι✝) → Measure (X✝ i)\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ns : Set ι\nhs : Countable ↑s\nt : (i : ι) → Set (X i... | [
"case ha\nι✝ : Type u_1\nX✝ : ι✝ → Type u_2\nmX✝ : (i : ι✝) → MeasurableSpace (X✝ i)\nμ✝ : (i : ι✝) → Measure (X✝ i)\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ns : Set ι\nhs : Countable ↑s\nt : (i : ι) → Set (X i)\n... | apply tendsto_nhds_unique (f := fun s' : Finset s ↦ ∏ i ∈ s', μ i (t i)) (l := atTop) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Probability.Kernel.Posterior | {
"line": 133,
"column": 33
} | {
"line": 133,
"column": 35
} | {
"line": 133,
"column": 36
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nthis : ⇑Kernel.id =ᵐ[μ] ⇑(Kernel.id†μ)\na : Ω\n⊢ Kernel.id a = (Kernel.id†μ) a → (Kernel.id†μ) a = Kernel.id a",
"ppTerm": "?m.86",
"assigned": true,
"usedCons... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nthis : ⇑Kernel.id =ᵐ[μ] ⇑(Kernel.id†μ)\na : Ω\nha : Kernel.id a = (Kernel.id†μ) a\n⊢ (Kernel.id†μ) a = Kernel.id a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Posterior | {
"line": 145,
"column": 8
} | {
"line": 145,
"column": 41
} | {
"line": 145,
"column": 41
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ Measure.map f μ ⊗ₘ (Kernel.deterministic f hf ∘ₖ ... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ Measure.map f μ ⊗ₘ (Kernel.deterministic f hf ∘ₖ (Kernel.dete... | Measure.deterministic_comp_eq_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 158,
"column": 6
} | {
"line": 158,
"column": 39
} | {
"line": 158,
"column": 39
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ ⇑(Kernel.copy 𝓧) ∘ₘ ⇑(Kernel.deterministic f hf)... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ ⇑(Kernel.copy 𝓧) ∘ₘ Measure.map f μ = ⇑(Kernel.copy 𝓧) ∘ₘ M... | Measure.deterministic_comp_eq_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 94
} | {
"line": 118,
"column": 2
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nhl : Measurable (uncurry ℓ)\nμ : Measure 𝓧\ninst✝¹ : SFinite μ\nπ : Measure Θ\ninst✝ : SFinite π\nhl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 → ∃ y, ∫⁻... | [
"Θ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nhl : Measurable (uncurry ℓ)\nμ : Measure 𝓧\ninst✝¹ : SFinite μ\nπ : Measure Θ\ninst✝ : SFinite π\nhl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 → ∃ y, ∫⁻ (θ : Θ), ℓ ... | rw [Measure.comp_apply_univ, ENNReal.iInf_mul' hl_pos (fun hμ ↦ h_zero (by simpa using hμ))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Decision.Risk.Countable | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 28
} | {
"line": 49,
"column": 29
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_3\n𝓨 : Type u_5\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nP : Kernel Θ 𝓧\nκ : Kernel 𝓧 𝓨\nπ : Measure Θ\ninst✝¹ : Fintype 𝓨\ninst✝ : MeasurableSingletonClass 𝓨\nhℓ : Measurable ℓ\n⊢ avgRisk ℓ P κ π = ∑ y, ∫⁻ (θ : Θ), ... | [
"Θ : Type u_1\n𝓧 : Type u_3\n𝓨 : Type u_5\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nP : Kernel Θ 𝓧\nκ : Kernel 𝓧 𝓨\nπ : Measure Θ\ninst✝¹ : Fintype 𝓨\ninst✝ : MeasurableSingletonClass 𝓨\nhℓ : Measurable ℓ\n⊢ ∑' (y : 𝓨), ∫⁻ (θ : Θ), ℓ θ y * (⇑κ ∘ₘ P θ) {y... | avgRisk_countable' hℓ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 316,
"column": 64
} | {
"line": 319,
"column": 6
} | {
"line": 321,
"column": 0
} | [
{
"pp": "𝓧 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\nπ : Measure Bool\ninst✝ : IsFiniteMeasure π\n⊢ ∀ᵐ (x : 𝓧) ∂⇑(Kernel.boolKernel μ ν) ∘ₘ π,\n (((Kernel.boolKernel μ ν)†π) x) {false} = π {false} * (∂μ/∂⇑(Kernel.boolKernel μ ν) ∘ₘ π) x"... | [] | by
filter_upwards [posterior_eq_withDensity_of_countable (Kernel.boolKernel μ ν) π] with x hx
rw [hx]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Posterior | {
"line": 324,
"column": 63
} | {
"line": 327,
"column": 6
} | {
"line": 329,
"column": 0
} | [
{
"pp": "𝓧 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\nπ : Measure Bool\ninst✝ : IsFiniteMeasure π\n⊢ ∀ᵐ (x : 𝓧) ∂⇑(Kernel.boolKernel μ ν) ∘ₘ π,\n (((Kernel.boolKernel μ ν)†π) x) {true} = π {true} * (∂ν/∂⇑(Kernel.boolKernel μ ν) ∘ₘ π) x",
... | [] | by
filter_upwards [posterior_eq_withDensity_of_countable (Kernel.boolKernel μ ν) π] with x hx
rw [hx]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Gamma | {
"line": 62,
"column": 90
} | {
"line": 63,
"column": 36
} | {
"line": 65,
"column": 0
} | [
{
"pp": "a r x : ℝ\nhx : 0 ≤ x\n⊢ gammaPDF a r x = ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)))",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"ProbabilityTheory.gammaPD... | [] | by
simp only [gammaPDF_eq, if_pos hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Binomial | {
"line": 114,
"column": 40
} | {
"line": 114,
"column": 73
} | {
"line": 116,
"column": 0
} | [
{
"pp": "n : ℕ\np : ↑I\n⊢ Bin(n, p).real {0} = (1 - ↑p) ^ n",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Nat.instOrderedSub",
"Real",
"ProbabilityTheory.binomial",
"Nat.choose",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrA... | [] | by simp [binomial_real_singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Binomial | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 22
} | {
"line": 119,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝³ : MeasurableSpace R\ninst✝² : AddMonoidWithOne R\ninst✝¹ : MeasurableSingletonClass R\ninst✝ : CharZero R\nn : ℕ\np : ↑I\n⊢ Bin(R, n, p).real {0} = (1 - ↑p) ^ n",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ProbabilityThe... | [
"R : Type u_1\ninst✝³ : MeasurableSpace R\ninst✝² : AddMonoidWithOne R\ninst✝¹ : MeasurableSingletonClass R\ninst✝ : CharZero R\nn : ℕ\np : ↑I\n⊢ Bin(R, n, p).real {↑0} = (1 - ↑p) ^ n"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Binomial | {
"line": 124,
"column": 34
} | {
"line": 124,
"column": 67
} | {
"line": 126,
"column": 0
} | [
{
"pp": "n : ℕ\np : ↑I\n⊢ Bin(n, p).real {n} = ↑p ^ n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"Nat.instOrderedSub",
"Real",
"ProbabilityTheory.binomial",
"Nat.choose",
"HMul.hMul",
"Mon... | [] | by simp [binomial_real_singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Exponential | {
"line": 138,
"column": 18
} | {
"line": 138,
"column": 20
} | {
"line": 138,
"column": 20
} | [
{
"pp": "r : ℝ\nhr : 0 < r\nx : ℝ\nh : 0 ≤ x\na : ℝ\n⊢ a ∈ Icc 0 x →\n (fun x ↦ ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)) a =\n (fun x ↦ ENNReal.ofReal (r * rexp (-(r * x)))) a",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
... | [
"r : ℝ\nhr : 0 < r\nx : ℝ\nh : 0 ≤ x\na : ℝ\nha : a ∈ Icc 0 x\n⊢ (fun x ↦ ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)) a =\n (fun x ↦ ENNReal.ofReal (r * rexp (-(r * x)))) a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 185,
"column": 46
} | {
"line": 187,
"column": 39
} | {
"line": 189,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddMonoidWithOne R\nmR : MeasurableSpace R\ninst✝ : MeasurableAdd₂ R\nr₁ r₂ : ℝ≥0\n⊢ Po(R, r₁) ∗ Po(R, r₂) = Po(R, r₁ + r₂)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"DiscreteUniformity.instCompleteS... | [] | by
rw [← Nat.coe_castAddMonoidHom, ← Measure.map_conv_addMonoidHom _ (by fun_prop),
poissonMeasure_conv_poissonMeasure] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 28
} | {
"line": 198,
"column": 0
} | [
{
"pp": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nr₁ r₂ : ℝ≥0\nX Y : Ω → ℕ\nhXY : X ⟂ᵢ[P] Y\nhX : HasLaw X Po(r₁) P\nhY : HasLaw Y Po(r₂) P\n⊢ HasLaw (X + Y) (Po(r₁) ∗ Po(r₂)) P",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompact... | [] | exact hXY.hasLaw_add hX hY | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 28
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddMonoidWithOne R\nmR : MeasurableSpace R\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝ : MeasurableAdd₂ R\nr₁ r₂ : ℝ≥0\nX Y : Ω → R\nhXY : X ⟂ᵢ[P] Y\nhX : HasLaw X Po(R, r₁) P\nhY : HasLaw Y Po(R, r₂) P\n⊢ HasLaw (X + Y) (Po(R, r₁) ∗ Po(R, r₂)) P",
"ppTerm": "... | [] | exact hXY.hasLaw_add hX hY | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.ZeroOne | {
"line": 40,
"column": 34
} | {
"line": 40,
"column": 36
} | {
"line": 41,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nt : Set Ω\nh_indep : ∀ᵐ (a : α) ∂μα, (κ a) (t ∩ t) = (κ a) t * (κ a) t\na : α\n⊢ (κ a) (t ∩ t) = (κ a) t * (κ a) t → (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ∞",
"ppTerm": "?m.55",
"assigne... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nt : Set Ω\nh_indep : ∀ᵐ (a : α) ∂μα, (κ a) (t ∩ t) = (κ a) t * (κ a) t\na : α\nha : (κ a) (t ∩ t) = (κ a) t * (κ a) t\n⊢ (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ∞"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.ZeroOne | {
"line": 227,
"column": 58
} | {
"line": 227,
"column": 60
} | {
"line": 227,
"column": 61
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nβ : Type u_4\np : Set ι → Prop\nf : Filter ι\nns : β → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nβ : Type u_4\np : Set ι → Prop\nf : Filter ι\nns : β → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.ZeroOne | {
"line": 285,
"column": 58
} | {
"line": 285,
"column": 60
} | {
"line": 285,
"column": 61
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nt : Set Ω\nht_tail : Measurabl... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nt : Set Ω\nht_tail : MeasurableSet t\na : ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.ZeroOne | {
"line": 342,
"column": 58
} | {
"line": 342,
"column": 60
} | {
"line": 342,
"column": 61
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nt : Set Ω\nht_tail : Measurabl... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nt : Set Ω\nht_tail : MeasurableSet t\na : ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 51
} | {
"line": 230,
"column": 4
} | [
{
"pp": "X Y : SFinKer\n⊢ Kernel.copy (X.carrier × Y.carrier) =\n Kernel.deterministic ⇑MeasurableEquiv.prodAssoc.symm ⋯ ∘ₖ\n (Kernel.id ∥ₖ Kernel.deterministic ⇑MeasurableEquiv.prodAssoc ⋯) ∘ₖ\n (Kernel.id ∥ₖ (Kernel.swap X.carrier Y.carrier ∥ₖ Kernel.id)) ∘ₖ\n (Kernel.id ∥ₖ... | [
"X Y : SFinKer\n⊢ Kernel.deterministic Function.diag ⋯ =\n Kernel.deterministic ⇑MeasurableEquiv.prodAssoc.symm ⋯ ∘ₖ\n (Kernel.deterministic id ⋯ ∥ₖ Kernel.deterministic ⇑MeasurableEquiv.prodAssoc ⋯) ∘ₖ\n (Kernel.deterministic id ⋯ ∥ₖ (Kernel.deterministic Prod.swap ⋯ ∥ₖ Kernel.determinis... | simp only [Kernel.copy, Kernel.id, Kernel.swap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 167,
"column": 6
} | {
"line": 168,
"column": 68
} | {
"line": 168,
"column": 68
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\nι : Type u_3\ninst✝¹⁰ : LinearOrder ι\ninst✝⁹ : LocallyFiniteOrder ι\ninst✝⁸ : OrderBot ι\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : DiscreteTopology ι\nin... | [
"Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\nι : Type u_3\ninst✝¹⁰ : LinearOrder ι\ninst✝⁹ : LocallyFiniteOrder ι\ninst✝⁸ : OrderBot ι\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : DiscreteTopology ι\ninst✝⁵ : Measu... | ae_eq_restrict_iff_indicator_ae_eq
(hτ.measurableSpace_le _ (hτ.measurableSet_le_stopping_time hσ)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 63,
"column": 14
} | {
"line": 63,
"column": 16
} | {
"line": 63,
"column": 17
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ₁ ρ₂ : Subrepresentation ρ\ng : G\n⊢ ∀ x₁ ∈ ρ₁.toSubmodule, ∀ x₂ ∈ ρ₂.toSubmodule, (... | [
"A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ₁ ρ₂ : Subrepresentation ρ\ng : G\nx₁ : W\n⊢ x₁ ∈ ρ₁.toSubmodule → ∀ x₂ ∈ ρ₂.toSubmodule, (ρ g) ... | x₁ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 44
} | {
"line": 70,
"column": 6
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ₁ ρ₂ : Subrepresentation ρ\n⊢ ∀ (g : G) ⦃v : W⦄, v ∈ ρ₁.toSubmodule ⊓ ρ₂.toSubmodule... | [
"A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ₁ ρ₂ : Subrepresentation ρ\n⊢ ∀ (g : G) ⦃v : W⦄, v ∈ ρ₁.toSubmodule → v ∈ ρ₂.toSubmodule → (ρ g)... | simp only [Submodule.mem_inf, and_imp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Intertwining | {
"line": 601,
"column": 35
} | {
"line": 601,
"column": 55
} | {
"line": 601,
"column": 56
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [
"A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ : Repre... | ← ρ.asAlgebraHom_of, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 39
} | {
"line": 125,
"column": 0
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nM : ModuleCat k[G]\n⊢ ↑((ofModuleMonoidAlgebra ⋙ toModuleMonoidAlgebra).obj M) ≃+ ↑M",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"RestrictScalars.module",
"Rep.ofModuleMo... | [] | dsimp [ofModuleMonoidAlgebra, toModuleMonoidAlgebra]
exact (Representation.ofModule M).asModuleEquiv.toAddEquiv.trans
(RestrictScalars.addEquiv k k[G] _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 39
} | {
"line": 125,
"column": 0
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nM : ModuleCat k[G]\n⊢ ↑((ofModuleMonoidAlgebra ⋙ toModuleMonoidAlgebra).obj M) ≃+ ↑M",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"RestrictScalars.module",
"Rep.ofModuleMo... | [] | dsimp [ofModuleMonoidAlgebra, toModuleMonoidAlgebra]
exact (Representation.ofModule M).asModuleEquiv.toAddEquiv.trans
(RestrictScalars.addEquiv k k[G] _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.StrongLaw | {
"line": 711,
"column": 2
} | {
"line": 712,
"column": 78
} | {
"line": 713,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X 0... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X 0)\nhindep : ... | let φ : ℕ → SimpleFunc E E :=
SimpleFunc.nearestPt (fun k => Nat.casesOn k 0 ((↑) ∘ denseSeq s) : ℕ → E) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.StrongLaw | {
"line": 756,
"column": 4
} | {
"line": 756,
"column": 19
} | {
"line": 757,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X 0... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X 0)\nhindep : ... | specialize hω k | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 80,
"column": 43
} | {
"line": 80,
"column": 67
} | {
"line": 80,
"column": 67
} | [
{
"pp": "case inr\nk : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ng₁ g₂ : G\na : ↑A\ns : ↥S\nh : ¬(QuotientGroup.rightRel S) g₂ g₁\n⊢ (A.indToCoindAux (↑s * g₁)) ((A.ρ s) a) g₂ = 0",
"ppTerm": "?inr",
... | [
"case inr\nk : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ng₁ g₂ : G\na : ↑A\ns : ↥S\nh : ¬(QuotientGroup.rightRel S) g₂ g₁\n⊢ 0 = 0",
"case inr.h\nk : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G... | indToCoindAux_of_not_rel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 88,
"column": 43
} | {
"line": 88,
"column": 67
} | {
"line": 88,
"column": 67
} | [
{
"pp": "case inr\nk : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ng₁ g₂ g₃ : G\na : ↑A\nh : ¬(QuotientGroup.rightRel S) (g₂ * g₃⁻¹) g₁\n⊢ 0 = (A.indToCoindAux (g₁ * g₃)) a g₂",
"ppTerm": "?inr",
"as... | [
"case inr\nk : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ng₁ g₂ g₃ : G\na : ↑A\nh : ¬(QuotientGroup.rightRel S) (g₂ * g₃⁻¹) g₁\n⊢ 0 = 0",
"case inr.h\nk : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Gro... | indToCoindAux_of_not_rel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 31
} | {
"line": 229,
"column": 6
} | [
{
"pp": "case succ\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nm : ℕ\na✝ : QuasiIsoAt (resolution.π k g) m\n⊢ (ModuleCat.Hom.hom\n ((HomologicalComplex.sc' ((chainComplexFunctor k g).obj (leftRegular k G)) (m + 2) (m + 1) m).... | [
"case pos\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nm : ℕ\na✝ : QuasiIsoAt (resolution.π k g) m\nhm : Odd (m + 1)\n⊢ (ModuleCat.Hom.hom\n ((HomologicalComplex.sc' ((chainComplexFunctor k g).obj (leftRegular k G)) (m + 2) (m + 1... | by_cases hm : Odd (m + 1) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 24
} | {
"line": 64,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\ni : ℕ\nx : leftRegular k G ⟶ A\n⊢ (ModuleCat.Hom.hom (A.leftRegularHomEquiv.toModuleIso.hom ≫ (moduleCatCochainComplex A g).d i (i + 1))) x =\n (ModuleCat.Hom.hom\n... | [
"case pos\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\ni : ℕ\nx : leftRegular k G ⟶ A\nhi : Even i\n⊢ (ModuleCat.Hom.hom (A.leftRegularHomEquiv.toModuleIso.hom ≫ (moduleCatCochainComplex A g).d i (i + 1))) x =\n (ModuleCa... | by_cases hi : Even i | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RepresentationTheory.Homological.Resolution | {
"line": 284,
"column": 2
} | {
"line": 288,
"column": 19
} | {
"line": 289,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nx✝ : ↑((standardComplex k G).X 1)\n⊢ (Hom.hom ((standardComplex k G).d 1 0 ≫ ε k G)).toLinearMap x✝ = (Hom.hom 0).toLinearMap x✝",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Rep.V",
"ChainComplex"... | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nx✝ : ↑((standardComplex k G).X 1)\nthis : (forget₂ToModuleCat k G).d 1 0 ≫ (forget₂ (Rep k G) (ModuleCat k)).map (ε k G) = 0\n⊢ (Hom.hom ((standardComplex k G).d 1 0 ≫ ε k G)).toLinearMap x✝ = (Hom.hom 0).toLinearMap x✝"
] | have : (forget₂ToModuleCat k G).d 1 0
≫ (forget₂ (Rep k G) (ModuleCat.{u} k)).map (ε k G) = 0 := by
rw [← forget₂ToModuleCatHomotopyEquiv_f_0_eq,
← (forget₂ToModuleCatHomotopyEquiv k G).1.2 1 0 rfl]
exact comp_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 659,
"column": 2
} | {
"line": 661,
"column": 45
} | {
"line": 663,
"column": 0
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : Group G\ninst✝¹ : CommGroup M\ninst✝ : MulAction G M\nf : G × G → M\nhf : IsMulCocycle₂ f\ng : G\n⊢ g • f (g⁻¹, g) / f (g, g⁻¹) = f (1, 1) / f (g, 1)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MulOne.toOne",
"... | [] | have := hf g g⁻¹ g
simp only [mul_inv_cancel, inv_mul_cancel, map_one_fst_of_isMulCocycle₂ hf g] at this
exact div_eq_div_iff_mul_eq_mul.2 this.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 659,
"column": 2
} | {
"line": 661,
"column": 45
} | {
"line": 663,
"column": 0
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : Group G\ninst✝¹ : CommGroup M\ninst✝ : MulAction G M\nf : G × G → M\nhf : IsMulCocycle₂ f\ng : G\n⊢ g • f (g⁻¹, g) / f (g, g⁻¹) = f (1, 1) / f (g, 1)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MulOne.toOne",
"... | [] | have := hf g g⁻¹ g
simp only [mul_inv_cancel, inv_mul_cancel, map_one_fst_of_isMulCocycle₂ hf g] at this
exact div_eq_div_iff_mul_eq_mul.2 this.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Ideal | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 31
} | {
"line": 160,
"column": 2
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nm : MonomialOrder σ\nB : Set (MvPolynomial σ R)\nh : 0 ∈ B\n⊢ span (m.leadingTerm '' insert 0 B) = span (m.leadingTerm '' B)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | [
"case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nm : MonomialOrder σ\nB : Set (MvPolynomial σ R)\nh : 0 ∉ B\n⊢ span (m.leadingTerm '' insert 0 B) = span (m.leadingTerm '' B)"
] | · rw [Set.insert_eq_of_mem h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 91,
"column": 37
} | {
"line": 93,
"column": 37
} | {
"line": 95,
"column": 0
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nf : σ → τ\ninst✝² : TendstoCofinite f\ninst✝¹ : DecidableEq σ\ninst✝ : DecidableEq τ\nx : τ →₀ ℕ\n⊢ Finset.image\n (fun x ↦\n match x with\n | (fst, b) => (b.1 + b.2, b))\n ⋯.toFinset =\n ⋯.toFinset",
"ppTerm": "?m.47",
"assigned": true,
... | [] | by
ext ⟨_, _, _⟩
simp; grind [Finsupp.mapDomain_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 134,
"column": 2
} | {
"line": 136,
"column": 23
} | {
"line": 138,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ Function.Surjective ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer A))",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Submodule",
"Representation.Coinvariants.ker",
"Rep.V",
"RingHomSurj... | [] | rintro ⟨x, hx⟩
rcases range_d₁₀_eq_coinvariantsKer A ▸ hx with ⟨y, hy⟩
use y, Subtype.ext hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 134,
"column": 2
} | {
"line": 136,
"column": 23
} | {
"line": 138,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ Function.Surjective ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer A))",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Submodule",
"Representation.Coinvariants.ker",
"Rep.V",
"RingHomSurj... | [] | rintro ⟨x, hx⟩
rcases range_d₁₀_eq_coinvariantsKer A ▸ hx with ⟨y, hy⟩
use y, Subtype.ext hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 187,
"column": 2
} | {
"line": 188,
"column": 6
} | {
"line": 190,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng h : G\na : ↑A\n⊢ (ConcreteCategory.hom (d₂₁ A)) (single (g⁻¹, g * h) ((A.ρ g⁻¹) a) + single (g, h) a) =\n single g⁻¹ ((A.ρ g⁻¹) a) + single g a",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | simp only [map_add, d₂₁_single (G := G), inv_inv, self_inv_apply, inv_mul_cancel_left]
abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 187,
"column": 2
} | {
"line": 188,
"column": 6
} | {
"line": 190,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng h : G\na : ↑A\n⊢ (ConcreteCategory.hom (d₂₁ A)) (single (g⁻¹, g * h) ((A.ρ g⁻¹) a) + single (g, h) a) =\n single g⁻¹ ((A.ρ g⁻¹) a) + single g a",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | simp only [map_add, d₂₁_single (G := G), inv_inv, self_inv_apply, inv_mul_cancel_left]
abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 690,
"column": 2
} | {
"line": 695,
"column": 5
} | {
"line": 697,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (shortComplexH0 A).Exact",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Representation.Coinvariants.ker",
"Rep.V",
"RingHomSurjective.ids",
"Finsupp.modu... | [] | rw [ShortComplex.moduleCat_exact_iff]
intro x (hx : Coinvariants.mk _ _ = 0)
rw [Coinvariants.mk_eq_zero, ← range_d₁₀_eq_coinvariantsKer] at hx
rcases hx with ⟨x, hx, rfl⟩
use x
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 690,
"column": 2
} | {
"line": 695,
"column": 5
} | {
"line": 697,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (shortComplexH0 A).Exact",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Representation.Coinvariants.ker",
"Rep.V",
"RingHomSurjective.ids",
"Finsupp.modu... | [] | rw [ShortComplex.moduleCat_exact_iff]
intro x (hx : Coinvariants.mk _ _ = 0)
rw [Coinvariants.mk_eq_zero, ← range_d₁₀_eq_coinvariantsKer] at hx
rcases hx with ⟨x, hx, rfl⟩
use x
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 62,
"column": 6
} | {
"line": 65,
"column": 94
} | {
"line": 66,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"le_max_right",
"Eq.mpr",
"Submodule",
"instHSMul",
"Semiring.toModule",
"instSMulOfMul",
... | [] | suffices ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j by
simpa only [smul_eq_mul, mul_top, Algebra.algebraMap_self, map_id, le_inf_iff] using! this
intro i j
exact ⟨max i j, pow_le_pow_right (le_max_left i j), pow_le_pow_right (le_max_right i j)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 62,
"column": 6
} | {
"line": 65,
"column": 94
} | {
"line": 66,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"le_max_right",
"Eq.mpr",
"Submodule",
"instHSMul",
"Semiring.toModule",
"instSMulOfMul",
... | [] | suffices ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j by
simpa only [smul_eq_mul, mul_top, Algebra.algebraMap_self, map_id, le_inf_iff] using! this
intro i j
exact ⟨max i j, pow_le_pow_right (le_max_left i j), pow_le_pow_right (le_max_right i j)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 50
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : CommRing R\nI : Ideal R⟦X⟧\nS : Set R\nP : Ideal R⟦X⟧\ninst✝² : P.IsPrime\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsDomain R\nw✝¹ : R\nw✝ : Set R⟦X⟧\nh₁ : span w✝ ≠ ⊥\nh₂ : (span w✝).IsPrime\nhXP : X ∉ span w✝\nh✝ : Ideal.map constantCoeff (span w✝) = R ∙ w✝¹\nleft✝ :... | [
"case neg\nR : Type u_1\ninst✝³ : CommRing R\nI : Ideal R⟦X⟧\nS : Set R\nP : Ideal R⟦X⟧\ninst✝² : P.IsPrime\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsDomain R\nw✝¹ : R\nw✝ : Set R⟦X⟧\nh₁ : span w✝ ≠ ⊥\nh₂ : (span w✝).IsPrime\nhXP : X ∉ span w✝\nh✝ : Ideal.map constantCoeff (span w✝) = R ∙ w✝¹\nleft✝ : w✝.Finite\n... | simp only [ncard_singleton, ncard_eq_one] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 34
} | {
"line": 269,
"column": 2
} | [
{
"pp": "case neg\nσ✝ : Type u_1\nR✝ : Type u_2\nn✝ : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nn : ℕ\nh : n ≠ 0\n⊢ (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) 1) = 1",
"ppTerm": "?neg✝",
"assigned": true,
... | [] | rw [truncTotal_one h, map_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 14
} | {
"line": 514,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\nA : Rep k G\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)\nx : G →₀ ↑A\nhxc : x ∈ cycles₁ A\nhx :\n (ConcreteCategory.hom (H1π (A.ofQuotient S)))\n ((ConcreteCategory.hom (mapCycles₁ (Quotie... | [
"k G : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\nA : Rep k G\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)\nx : G →₀ ↑A\nhxc : x ∈ cycles₁ A\nhx :\n (ConcreteCategory.hom (H1π (A.ofQuotient S)))\n ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' ... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 573,
"column": 2
} | {
"line": 574,
"column": 61
} | {
"line": 575,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂\na✝ : x ∈ ⊤\nX : G →₀ ↥S →₀ ↑A\nhX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x\nY : ↥S →₀ ↑A :=\n X... | [
"k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂\na✝ : x ∈ ⊤\nX : G →₀ ↥S →₀ ↑A\nhX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x\nY : ↥S →₀ ↑A :=\n X.sum fun g f... | change d₂₁ A Z = mapRange id rfl (lmapDomain _ k Subtype.val Y) -
mapRange.linearMap (Submodule.subtype _) (mapDomain id x) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
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