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.Category.SFinKer | {
"line": 170,
"column": 55
} | {
"line": 170,
"column": 57
} | {
"line": 170,
"column": 58
} | [
{
"pp": "X Y : SFinKer\n⊢ Measurable Prod.snd",
"ppTerm": "?m.1125",
"assigned": true,
"usedConstants": [
"SFinKer.of",
"SFinKer.str",
"PUnit",
"SFinKer.carrier",
"PUnit.instMeasurableSpace",
"measurable_id'",
"Prod",
"Measurable.snd",
"Prod.inst... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 170,
"column": 41
} | {
"line": 170,
"column": 43
} | {
"line": 170,
"column": 44
} | [
{
"pp": "X Y : SFinKer\n⊢ Measurable id",
"ppTerm": "?m.1146",
"assigned": true,
"usedConstants": [
"SFinKer.of",
"SFinKer.str",
"PUnit",
"SFinKer.carrier",
"PUnit.instMeasurableSpace",
"measurable_id'",
"Prod",
"Prod.instMeasurableSpace"
],
"u... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 170,
"column": 55
} | {
"line": 170,
"column": 57
} | {
"line": 170,
"column": 58
} | [
{
"pp": "X Y : SFinKer\n⊢ Measurable Prod.fst",
"ppTerm": "?m.1148",
"assigned": true,
"usedConstants": [
"SFinKer.of",
"Measurable.fst",
"SFinKer.str",
"PUnit",
"SFinKer.carrier",
"PUnit.instMeasurableSpace",
"measurable_id'",
"Prod",
"Prod.inst... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 171,
"column": 64
} | {
"line": 171,
"column": 66
} | {
"line": 171,
"column": 67
} | [
{
"pp": "X Y : SFinKer\n⊢ Measurable id",
"ppTerm": "?m.1172",
"assigned": true,
"usedConstants": [
"SFinKer.str",
"SFinKer.carrier",
"measurable_id'"
],
"usedFVars": [
"X"
],
"usedGoals": []
}
] | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 171,
"column": 78
} | {
"line": 171,
"column": 80
} | {
"line": 171,
"column": 81
} | [
{
"pp": "X Y : SFinKer\n⊢ Measurable (Prod.snd ∘ id)",
"ppTerm": "?m.1173",
"assigned": true,
"usedConstants": [
"SFinKer.of",
"id",
"SFinKer.str",
"PUnit",
"SFinKer.carrier",
"PUnit.instMeasurableSpace",
"measurable_id'",
"Prod",
"Measurable.snd... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 171,
"column": 64
} | {
"line": 171,
"column": 66
} | {
"line": 171,
"column": 67
} | [
{
"pp": "X Y : SFinKer\n⊢ Measurable (Prod.fst ∘ id)",
"ppTerm": "?m.1197",
"assigned": true,
"usedConstants": [
"SFinKer.of",
"id",
"Measurable.fst",
"SFinKer.str",
"PUnit",
"SFinKer.carrier",
"PUnit.instMeasurableSpace",
"measurable_id'",
"Prod... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 171,
"column": 78
} | {
"line": 171,
"column": 80
} | {
"line": 171,
"column": 81
} | [
{
"pp": "X Y : SFinKer\n⊢ Measurable id",
"ppTerm": "?m.1198",
"assigned": true,
"usedConstants": [
"SFinKer.str",
"SFinKer.carrier",
"measurable_id'"
],
"usedFVars": [
"Y"
],
"usedGoals": []
}
] | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 167,
"column": 18
} | {
"line": 167,
"column": 20
} | {
"line": 168,
"column": 4
} | [
{
"pp": "X Y : SFinKer\n⊢ { hom := { hom := Kernel.deterministic ⇑prodAssoc ⋯, property := ⋯ },\n inv := { hom := Kernel.deterministic ⇑prodAssoc.symm ⋯, property := ⋯ }, hom_inv_id := ⋯,\n inv_hom_id := ⋯ }.hom ≫\n {\n hom :=\n Kernel.id ∥ₖ\n (let f₁ := fun x ↦... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 664,
"column": 97
} | {
"line": 664,
"column": 99
} | {
"line": 665,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : Measurable f\nhg : Measurable g\n⊢ CondIndepFun m' hm' f g μ ↔\n ... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 178,
"column": 30
} | {
"line": 178,
"column": 32
} | {
"line": 178,
"column": 33
} | [
{
"pp": "X Y : SFinKer\n⊢ IsSFiniteKernel (Kernel.swap X.carrier Y.carrier)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"ProbabilityTheory.IsZeroOrMarkovKernel.isFiniteKernel",
"congrArg",
"ProbabilityTheory.IsSFin... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 179,
"column": 24
} | {
"line": 179,
"column": 26
} | {
"line": 179,
"column": 27
} | [
{
"pp": "X Y : SFinKer\n⊢ IsSFiniteKernel (Kernel.swap Y.carrier X.carrier)",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"ProbabilityTheory.IsZeroOrMarkovKernel.isFiniteKernel",
"congrArg",
"ProbabilityTheory.IsSFin... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 177,
"column": 18
} | {
"line": 177,
"column": 20
} | {
"line": 178,
"column": 4
} | [
{
"pp": "X Y : SFinKer\n⊢ X ⊗ Y ≅ Y ⊗ X",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"SFinKer.Hom.hom",
"ProbabilityTheory.Kernel.swap_swap",
"congrArg",
"SFinKer",
"CategoryTheory.CategoryStruct... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 182,
"column": 39
} | {
"line": 182,
"column": 41
} | {
"line": 183,
"column": 4
} | [
{
"pp": "X Y Z : SFinKer\nκ : Y ⟶ Z\n⊢ X ◁ κ ≫\n { hom := { hom := Kernel.swap X.carrier Z.carrier, property := ⋯ },\n inv := { hom := Kernel.swap Z.carrier X.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom =\n { hom := { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ },... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 185,
"column": 34
} | {
"line": 185,
"column": 36
} | {
"line": 186,
"column": 4
} | [
{
"pp": "X✝ Y✝ : SFinKer\nκ : X✝ ⟶ Y✝\nX : SFinKer\n⊢ κ ▷ X ≫\n { hom := { hom := Kernel.swap Y✝.carrier X.carrier, property := ⋯ },\n inv := { hom := Kernel.swap X.carrier Y✝.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom =\n { hom := { hom := Kernel.swap X✝.carrier X.carrier... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 188,
"column": 27
} | {
"line": 188,
"column": 29
} | {
"line": 189,
"column": 4
} | [
{
"pp": "X Y Z : SFinKer\n⊢ (α_ X Y Z).hom ≫\n { hom := { hom := Kernel.swap X.carrier (Y ⊗ Z).carrier, property := ⋯ },\n inv := { hom := Kernel.swap (Y ⊗ Z).carrier X.carrier, property := ⋯ }, hom_inv_id := ⋯,\n inv_hom_id := ⋯ }.hom ≫\n (α_ Y Z X).hom =\n { hom := { hom :... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 687,
"column": 49
} | {
"line": 687,
"column": 51
} | {
"line": 688,
"column": 8
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nβ : ι → Type u_5\nm : (x : ι) → MeasurableSpace (β x)\nf : (i : ι) → Ω → β i\nhf : ∀ (i : ι), Measurable (f i)\ns : Finset ι\nsets : ι → Set Ω\nh_sets : ∀ i ∈ s... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.Conditional | {
"line": 675,
"column": 85
} | {
"line": 675,
"column": 87
} | {
"line": 676,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nβ : ι → Type u_5\nm : (x : ι) → MeasurableSpace (β x)\nf : (i : ι) → Ω → β i\nhf : ∀ (i : ι), Measurable (f i)\n⊢ iCondIndepFun m' hm' f μ ↔\n ∀ (S : Finset ... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 696,
"column": 94
} | {
"line": 696,
"column": 96
} | {
"line": 697,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : Measurable f\nhg : Measurable g\n⊢ CondIndepFun m' hm' f g μ ↔\n ... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Proper | {
"line": 126,
"column": 58
} | {
"line": 126,
"column": 60
} | {
"line": 127,
"column": 2
} | [
{
"pp": "X : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nf g : X → ℝ≥0∞\nhπ : π.IsProper\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nhf : Measurable f\nhg : Measurable g\nx₀ : X\n⊢ ∫⁻ (x : X), g x * f x ∂π x₀ = g x₀ * ∫⁻ (x : X), f x ∂π x₀",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ENNReal.inst... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 51,
"column": 38
} | {
"line": 51,
"column": 40
} | {
"line": 51,
"column": 41
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\na : ↑I\ns : X\nx y : ↑I\nhxy : x ≤ y\n⊢ Icc 0 x ⊆ Icc 0 y",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRi... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 194,
"column": 27
} | {
"line": 194,
"column": 29
} | {
"line": 195,
"column": 4
} | [
{
"pp": "X Y Z : SFinKer\n⊢ (α_ X Y Z).inv ≫\n { hom := { hom := Kernel.swap (X ⊗ Y).carrier Z.carrier, property := ⋯ },\n inv := { hom := Kernel.swap Z.carrier (X ⊗ Y).carrier, property := ⋯ }, hom_inv_id := ⋯,\n inv_hom_id := ⋯ }.hom ≫\n (α_ Z X Y).inv =\n X ◁\n { h... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 736,
"column": 67
} | {
"line": 736,
"column": 69
} | {
"line": 737,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nX : Ω → β\nY : Ω → β'\nhX : Measurable X\nhY : Measurable Y\ns : Set β\nhs : MeasurableSet s\nt... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 200,
"column": 18
} | {
"line": 200,
"column": 20
} | {
"line": 201,
"column": 4
} | [
{
"pp": "X Y : SFinKer\n⊢ { hom := { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ },\n inv := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n { hom := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ },\n inv := { hom :... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 200,
"column": 18
} | {
"line": 201,
"column": 17
} | {
"line": 203,
"column": 0
} | [
{
"pp": "X Y : SFinKer\n⊢ { hom := { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ },\n inv := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n { hom := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ },\n inv := { hom :... | [] | by
ext : 1; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 205,
"column": 31
} | {
"line": 205,
"column": 33
} | {
"line": 205,
"column": 34
} | [
{
"pp": "X : SFinKer\n⊢ IsSFiniteKernel (Kernel.discard X.carrier)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ProbabilityTheory.Kernel.discard.eq_1",
"SFinKer.instMonoidalCategory",
"ProbabilityTheory.IsZeroOrMarkovKernel.isFiniteKernel",
"cong... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 206,
"column": 27
} | {
"line": 206,
"column": 29
} | {
"line": 206,
"column": 30
} | [
{
"pp": "X : SFinKer\n⊢ IsSFiniteKernel (Kernel.copy X.carrier)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"ProbabilityTheory.IsZeroOrMarkovKernel.isFiniteKernel",
"ProbabilityTheory.Kernel.copy.eq_1",
"congrArg",... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 211,
"column": 87
} | {
"line": 211,
"column": 89
} | {
"line": 211,
"column": 90
} | [
{
"pp": "X : SFinKer\n⊢ Measurable fun x ↦ (PUnit.unit, x)",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"SFinKer.of",
"Measurable.prodMk",
"SFinKer.str",
"PUnit",
"SFinKer.carrier",
"PUnit.instMeasurableSpace",
"measurable_id'",
"measurabl... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 207,
"column": 18
} | {
"line": 207,
"column": 20
} | {
"line": 208,
"column": 4
} | [
{
"pp": "X : SFinKer\n⊢ { hom := Kernel.copy X.carrier, property := ⋯ } ≫ { hom := Kernel.discard X.carrier, property := ⋯ } ▷ X = (λ_ X).inv",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.Kernel.deterministic_map",
"Eq.mpr",
"SFinKer.instMonoidalCateg... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 217,
"column": 87
} | {
"line": 217,
"column": 89
} | {
"line": 217,
"column": 90
} | [
{
"pp": "X : SFinKer\n⊢ Measurable fun x ↦ (x, PUnit.unit)",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"SFinKer.of",
"Measurable.prodMk",
"SFinKer.str",
"PUnit",
"SFinKer.carrier",
"PUnit.instMeasurableSpace",
"measurable_id'",
"measurab... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 213,
"column": 18
} | {
"line": 213,
"column": 20
} | {
"line": 214,
"column": 4
} | [
{
"pp": "X : SFinKer\n⊢ { hom := Kernel.copy X.carrier, property := ⋯ } ≫ X ◁ { hom := Kernel.discard X.carrier, property := ⋯ } = (ρ_ X).inv",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.Kernel.deterministic_map",
"Eq.mpr",
"SFinKer.instMonoidalCateg... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 741,
"column": 75
} | {
"line": 741,
"column": 77
} | {
"line": 742,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nX : Ω → β\nY : Ω → β'\nhX : Measurable X\nhY : Measurable Y\ns : Set β\nhs : MeasurableSet s\nt... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 219,
"column": 17
} | {
"line": 219,
"column": 19
} | {
"line": 220,
"column": 4
} | [
{
"pp": "X : SFinKer\n⊢ { hom := Kernel.copy X.carrier, property := ⋯ } ≫ X ◁ { hom := Kernel.copy X.carrier, property := ⋯ } =\n { hom := Kernel.copy X.carrier, property := ⋯ } ≫\n { hom := Kernel.copy X.carrier, property := ⋯ } ▷ X ≫ (α_ X X X).hom",
"ppTerm": "?m.153",
"assigned": true,
"... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 184,
"column": 41
} | {
"line": 184,
"column": 43
} | {
"line": 184,
"column": 44
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nG : Type u_6\nH : Type u_7\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : HasOuterApproxClosed G\ninst✝⁴ : TopologicalSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : BorelSpace H\ninst✝¹ : HasOuterApproxClosed... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 731,
"column": 33
} | {
"line": 731,
"column": 35
} | {
"line": 732,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nX : Ω → β\nY : Ω → β'\nhX : Measurable X\nhY : Measurable Y\n⊢ CondIndepFun m' hm' X Y μ",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 226,
"column": 23
} | {
"line": 226,
"column": 25
} | {
"line": 226,
"column": 26
} | [
{
"pp": "X : SFinKer\n⊢ Δ ≫ (β_ X X).hom = Δ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"SFinKer.Hom.hom",
"SFinKer.instSymmetricCategory",
"SFinKer",
"id",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 731,
"column": 33
} | {
"line": 744,
"column": 76
} | {
"line": 746,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nX : Ω → β\nY : Ω → β'\nhX : Measurable X\nhY : Measurable Y\n⊢ CondIndepFun m' hm' X Y μ",
... | [] | by
rw [condIndepFun_iff _ hm' _ _ (hX.mono hm' le_rfl) hY]
rintro _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
rw [show (fun ω : Ω ↦ (1 : ℝ)) = 1 from rfl, Set.inter_indicator_one]
calc μ[(X ⁻¹' s).indicator 1 * (Y ⁻¹' t).indicator 1 | m']
_ =ᵐ[μ] (X ⁻¹' s).indicator 1 * μ[(Y ⁻¹' t).indicator 1 | m'] := by
refine condEx... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 178,
"column": 22
} | {
"line": 178,
"column": 24
} | {
"line": 179,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nG : Type u_6\nH : Type u_7\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : HasOuterApproxClosed G\ninst✝⁴ : TopologicalSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : BorelSpace H\ninst✝¹ : HasOuterApproxClosed... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 790,
"column": 83
} | {
"line": 790,
"column": 85
} | {
"line": 791,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝¹ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝ : CountableOrCountablyGenerated Ω (β × β')\nhf : Measurable f\nhg... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 64,
"column": 20
} | {
"line": 64,
"column": 22
} | {
"line": 64,
"column": 23
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\na : ↑I\nh_monotone : ∀ (s : X), Monotone fun x ↦ (κ s).real (Icc 0 x)\ne : X × ↑I\nq : ℚ\nhqa : ↑a < ↑q\nhqI : 0 ≤ q ∧ ↑q ≤ 1\nh : (κ e.1).real (Icc 0 ⟨↑q,... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 201,
"column": 81
} | {
"line": 201,
"column": 83
} | {
"line": 202,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Fintype S\ninst... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.Conditional | {
"line": 807,
"column": 30
} | {
"line": 807,
"column": 32
} | {
"line": 807,
"column": 33
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : Measurable f\nhg : Measurable g\n⊢ Measurable fun ω ↦ (f ω, g ω)",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 808,
"column": 27
} | {
"line": 808,
"column": 29
} | {
"line": 808,
"column": 30
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : Measurable f\nhg : Measurable g\n⊢ Measurable (Prod.map id fun ω ↦ ... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 804,
"column": 27
} | {
"line": 804,
"column": 29
} | {
"line": 805,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : Measurable f\nhg : Measurable g\n⊢ CondIndepFun m' hm' f g μ ↔\n ... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 53,
"column": 97
} | {
"line": 53,
"column": 99
} | {
"line": 54,
"column": 6
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\na : ↑I\nh_monotone : ∀ (s : X), Monotone fun x ↦ (κ s).real (Icc 0 x)\n⊢ {x | a < f x.1 x.2} = ⋃ q, ⋃ (hqI : ↑q ∈ I), ⋃ (_ : ↑a < ↑q), {e | (κ e.1).real (I... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 206,
"column": 68
} | {
"line": 206,
"column": 70
} | {
"line": 207,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Fintype S\ninst... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.Conditional | {
"line": 830,
"column": 27
} | {
"line": 830,
"column": 29
} | {
"line": 830,
"column": 30
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 833,
"column": 60
} | {
"line": 833,
"column": 62
} | {
"line": 833,
"column": 63
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 227,
"column": 21
} | {
"line": 227,
"column": 23
} | {
"line": 228,
"column": 4
} | [
{
"pp": "X Y : SFinKer\n⊢ Δ = (Δ ⊗ₘ Δ) ≫ MonoidalCategory.tensorμ X X Y Y",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"SFin... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 47,
"column": 48
} | {
"line": 47,
"column": 50
} | {
"line": 48,
"column": 4
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\n⊢ Measurable (uncurry f)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Rat.instOfNat",
"Set.ext",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.Conditional | {
"line": 829,
"column": 62
} | {
"line": 829,
"column": 64
} | {
"line": 830,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 236,
"column": 23
} | {
"line": 236,
"column": 25
} | {
"line": 236,
"column": 26
} | [
{
"pp": "X Y : SFinKer\n⊢ Measurable Prod.snd",
"ppTerm": "?m.200",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer",
"SFinKer.of",
"SFinKer.str",
"PUnit",
"SFinKer.carrier",
"CategoryTheory.MonoidalCategoryStruct.tensorUnit",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 238,
"column": 28
} | {
"line": 238,
"column": 30
} | {
"line": 238,
"column": 31
} | [
{
"pp": "X Y : SFinKer\na✝¹ : X.carrier × Y.carrier\ns✝ : Set PUnit.{u + 1}\na✝ : MeasurableSet s✝\n⊢ Measurable Prod.snd",
"ppTerm": "?m.246",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer",
"SFinKer.str",
"PUnit",
"CategoryTheory.Monoid... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 233,
"column": 24
} | {
"line": 233,
"column": 26
} | {
"line": 234,
"column": 4
} | [
{
"pp": "X Y : SFinKer\n⊢ ε = (ε ⊗ₘ ε) ≫ (λ_ (𝟙_ SFinKer)).hom",
"ppTerm": "?m.187",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"ProbabilityTheory.Kernel.map_apply",
"ProbabilityTheory.IsZeroOrMarkovKernel.isFinite... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 75,
"column": 39
} | {
"line": 75,
"column": 41
} | {
"line": 75,
"column": 42
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\nmeasurable_f : Measurable (uncurry f)\na : X\nx : ↑I\n⊢ Iic x = Icc 0 x",
"ppTerm": "?m.433",
"assigned": true,
"usedConstants": [
"Set.e... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 243,
"column": 23
} | {
"line": 243,
"column": 25
} | {
"line": 243,
"column": 26
} | [
{
"pp": "a✝¹ : PUnit.{u + 1}\ns✝ : Set (PUnit.{u + 1} × PUnit.{u + 1})\na✝ : MeasurableSet s✝\n⊢ Measurable fun x ↦ (PUnit.unit, x)",
"ppTerm": "?m.301",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer",
"SFinKer.of",
"Measurable.prodMk",
"... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 240,
"column": 15
} | {
"line": 240,
"column": 17
} | {
"line": 241,
"column": 4
} | [
{
"pp": "⊢ Δ = (λ_ (𝟙_ SFinKer)).inv",
"ppTerm": "?m.273",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"MeasureTheory.Measure",
"SFinKer.Hom.hom",
"instCountablePUnit",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 840,
"column": 61
} | {
"line": 840,
"column": 63
} | {
"line": 840,
"column": 64
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 249,
"column": 15
} | {
"line": 249,
"column": 17
} | {
"line": 250,
"column": 4
} | [
{
"pp": "X Y : SFinKer\nκ : Kernel X.carrier Y.carrier\ninst✝¹ : IsDeterministic κ\ninst✝ : IsMarkovKernel κ\n⊢ { hom := κ, property := ⋯ } ≫ Δ = Δ ≫ ({ hom := κ, property := ⋯ } ⊗ₘ { hom := κ, property := ⋯ })",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 256,
"column": 15
} | {
"line": 256,
"column": 17
} | {
"line": 256,
"column": 18
} | [
{
"pp": "X Y : SFinKer\nf : X.carrier → Y.carrier\nhf : Measurable f\n⊢ { hom := Kernel.id.map f, property := ⋯ } ≫ Δ =\n Δ ≫ ({ hom := Kernel.id.map f, property := ⋯ } ⊗ₘ { hom := Kernel.id.map f, property := ⋯ })",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"CategoryTheory.Co... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 265,
"column": 52
} | {
"line": 265,
"column": 54
} | {
"line": 265,
"column": 55
} | [
{
"pp": "X : SFinKer\nY : SFinKer\nZ : SFinKer\n⊢ Measurable Prod.snd",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer",
"SFinKer.str",
"CategoryTheory.MonoidalCategoryStruct.tensorObj",
"SFinKer.carrier",
"Categor... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 268,
"column": 52
} | {
"line": 268,
"column": 54
} | {
"line": 268,
"column": 55
} | [
{
"pp": "X : SFinKer\nY : SFinKer\nZ : SFinKer\n⊢ Measurable Prod.fst",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer",
"Measurable.fst",
"SFinKer.str",
"CategoryTheory.MonoidalCategoryStruct.tensorObj",
"SFinKer.... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 271,
"column": 79
} | {
"line": 271,
"column": 81
} | {
"line": 271,
"column": 82
} | [
{
"pp": "X Y : SFinKer\nZ : SFinKer\n⊢ Measurable Prod.swap",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"SFinKer.str",
"measurable_swap",
"SFinKer.carrier"
],
"usedFVars": [
"X",
"Y"
],
"usedGoals": []
}
] | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 275,
"column": 54
} | {
"line": 275,
"column": 56
} | {
"line": 275,
"column": 57
} | [
{
"pp": "X : SFinKer\nY : SFinKer\nZ : SFinKer\n⊢ Measurable fun x ↦ PUnit.unit",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer",
"SFinKer.str",
"SFinKer.carrier",
"CategoryTheory.MonoidalCategoryStruct.tensorUnit",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 278,
"column": 86
} | {
"line": 278,
"column": 88
} | {
"line": 278,
"column": 89
} | [
{
"pp": "X : SFinKer\nY : SFinKer\nZ : SFinKer\n⊢ Measurable fun x ↦ (x, x)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Measurable.prodMk",
"SFinKer.str",
"SFinKer.carrier",
"measurable_id'"
],
"usedFVars": [
"X"
],
"usedGoals": []
}
] | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 841,
"column": 49
} | {
"line": 841,
"column": 51
} | {
"line": 841,
"column": 52
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 55,
"column": 68
} | {
"line": 55,
"column": 70
} | {
"line": 56,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 841,
"column": 81
} | {
"line": 841,
"column": 83
} | {
"line": 841,
"column": 84
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 210,
"column": 67
} | {
"line": 210,
"column": 69
} | {
"line": 211,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Fintype S\ninst... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 70,
"column": 35
} | {
"line": 70,
"column": 37
} | {
"line": 70,
"column": 38
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 64,
"column": 68
} | {
"line": 64,
"column": 70
} | {
"line": 65,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 86,
"column": 34
} | {
"line": 86,
"column": 36
} | {
"line": 87,
"column": 6
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\nmeasurable_f : Measurable (uncurry f)\na : X\nx : ↑I\nIic_to_Icc : Iic x = Icc 0 x\nκ_in_I : (κ a).real (Icc 0 x) ∈ I\nξ : ↑I\nhξ : f a ξ ≤ x\nhx : ¬x = 1\... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 80,
"column": 81
} | {
"line": 80,
"column": 83
} | {
"line": 81,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 94,
"column": 56
} | {
"line": 94,
"column": 58
} | {
"line": 95,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 101,
"column": 32
} | {
"line": 101,
"column": 34
} | {
"line": 102,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 93,
"column": 58
} | {
"line": 93,
"column": 60
} | {
"line": 94,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 125,
"column": 71
} | {
"line": 125,
"column": 73
} | {
"line": 126,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 94,
"column": 70
} | {
"line": 94,
"column": 72
} | {
"line": 94,
"column": 73
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\nmeasurable_f : Measurable (uncurry f)\na : X\nx : ↑I\nIic_to_Icc : Iic x = Icc 0 x\nκ_in_I : (κ a).real (Icc 0 x) ∈ I\nξ : ↑I\nhξ : f a ξ ≤ x\nhx : ¬x = 1\... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 172,
"column": 90
} | {
"line": 172,
"column": 92
} | {
"line": 173,
"column": 4
} | [
{
"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... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.Conditional | {
"line": 839,
"column": 53
} | {
"line": 839,
"column": 55
} | {
"line": 840,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 176,
"column": 79
} | {
"line": 176,
"column": 81
} | {
"line": 177,
"column": 4
} | [
{
"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... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 166,
"column": 26
} | {
"line": 166,
"column": 28
} | {
"line": 167,
"column": 2
} | [
{
"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... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 45,
"column": 79
} | {
"line": 45,
"column": 81
} | {
"line": 46,
"column": 2
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\n⊢ ∃ f, Measurable (uncurry f) ∧ ∀ (a : X), Measure.map (f a) volume = κ a",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"Rat.instOfNat",
"Set.ext",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 115,
"column": 42
} | {
"line": 115,
"column": 44
} | {
"line": 115,
"column": 45
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nmX : MeasurableSpace X\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nκ : Kernel X Y\ninst✝ : IsMarkovKernel κ\ng : Y → ↑I := sigmoid ∘ embeddingReal Y\nhg : MeasurableEmbedding (sigmoid ∘ embeddingReal Y)\nhκg : IsMarkovKernel (κ.map g)\n⊢ Meas... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 114,
"column": 45
} | {
"line": 114,
"column": 47
} | {
"line": 115,
"column": 4
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nmX : MeasurableSpace X\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nκ : Kernel X Y\ninst✝ : IsMarkovKernel κ\ng : Y → ↑I := sigmoid ∘ embeddingReal Y\nhg : MeasurableEmbedding (sigmoid ∘ embeddingReal Y)\nhκg : IsMarkovKernel (κ.map g)\n⊢ κ = ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Kernel.Representation | {
"line": 118,
"column": 40
} | {
"line": 118,
"column": 42
} | {
"line": 118,
"column": 43
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nmX : MeasurableSpace X\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nκ : Kernel X Y\ninst✝ : IsMarkovKernel κ\ng : Y → ↑I := sigmoid ∘ embeddingReal Y\nhg : MeasurableEmbedding (sigmoid ∘ embeddingReal Y)\nhκg : IsMarkovKernel (κ.map g)\nhg'κ :... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 119,
"column": 25
} | {
"line": 119,
"column": 27
} | {
"line": 119,
"column": 28
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nmX : MeasurableSpace X\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nκ : Kernel X Y\ninst✝ : IsMarkovKernel κ\ng : Y → ↑I := sigmoid ∘ embeddingReal Y\nhg : MeasurableEmbedding (sigmoid ∘ embeddingReal Y)\nhκg : IsMarkovKernel (κ.map g)\nhg'κ :... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 119,
"column": 64
} | {
"line": 119,
"column": 66
} | {
"line": 119,
"column": 67
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nmX : MeasurableSpace X\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nκ : Kernel X Y\ninst✝ : IsMarkovKernel κ\ng : Y → ↑I := sigmoid ∘ embeddingReal Y\nhg : MeasurableEmbedding (sigmoid ∘ embeddingReal Y)\nhκg : IsMarkovKernel (κ.map g)\nhg'κ :... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 119,
"column": 78
} | {
"line": 119,
"column": 80
} | {
"line": 119,
"column": 81
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nmX : MeasurableSpace X\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nκ : Kernel X Y\ninst✝ : IsMarkovKernel κ\ng : Y → ↑I := sigmoid ∘ embeddingReal Y\nhg : MeasurableEmbedding (sigmoid ∘ embeddingReal Y)\nhκg : IsMarkovKernel (κ.map g)\nhg'κ :... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 110,
"column": 79
} | {
"line": 110,
"column": 81
} | {
"line": 111,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nmX : MeasurableSpace X\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nκ : Kernel X Y\ninst✝ : IsMarkovKernel κ\n⊢ ∃ f, Measurable (uncurry f) ∧ ∀ (a : X), Measure.map (f a) volume = κ a",
"ppTerm": "?m.26",
"assigned": true,
"usedCon... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 128,
"column": 15
} | {
"line": 128,
"column": 17
} | {
"line": 128,
"column": 18
} | [
{
"pp": "Y : Type u_2\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nμ : Measure Y\ninst✝ : IsProbabilityMeasure μ\nf : Unit → ↑I → Y\nhf_meas : Measurable (uncurry f)\nhf_map : map (f ()) volume = (Kernel.const Unit μ) ()\n⊢ Measurable (f ())",
"ppTerm": "?m.54",
"assigned... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 128,
"column": 28
} | {
"line": 128,
"column": 30
} | {
"line": 128,
"column": 31
} | [
{
"pp": "Y : Type u_2\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nμ : Measure Y\ninst✝ : IsProbabilityMeasure μ\nf : Unit → ↑I → Y\nhf_meas : Measurable (uncurry f)\nhf_map : map (f ()) volume = (Kernel.const Unit μ) ()\n⊢ map (f ()) volume = μ",
"ppTerm": "?m.55",
"assi... | [] | by | [anonymous] | by |
Mathlib.Probability.Kernel.Representation | {
"line": 125,
"column": 54
} | {
"line": 125,
"column": 56
} | {
"line": 126,
"column": 2
} | [
{
"pp": "Y : Type u_2\ninst✝² : Nonempty Y\nmY : MeasurableSpace Y\ninst✝¹ : StandardBorelSpace Y\nμ : Measure Y\ninst✝ : IsProbabilityMeasure μ\n⊢ ∃ f, Measurable f ∧ map f volume = μ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Unit.unit",
"Real",
"MeasureTheory.Mea... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 45,
"column": 87
} | {
"line": 45,
"column": 89
} | {
"line": 46,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\ns : Set Ω\nhs : MeasurableSet s\n⊢ (μ.tilted fun x ↦ t * X x) s = ∫⁻ (a : Ω) in s, ENNReal.ofReal (rexp (t * X a) / mgf X μ t) ∂μ",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProd... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 49,
"column": 87
} | {
"line": 49,
"column": 89
} | {
"line": 50,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\ninst✝ : SFinite μ\ns : Set Ω\n⊢ (μ.tilted fun x ↦ t * X x) s = ∫⁻ (a : Ω) in s, ENNReal.ofReal (rexp (t * X a) / mgf X μ t) ∂μ",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProduct... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 54,
"column": 87
} | {
"line": 54,
"column": 89
} | {
"line": 55,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\ns : Set Ω\nhs : MeasurableSet s\nht : Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ (μ.tilted fun x ↦ t * X x) s = ∫⁻ (a : Ω) in s, ENNReal.ofReal (rexp (t * X a - cgf X μ t)) ∂μ",
"ppTerm": "?m.39",
"assigned": true,
"usedCo... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 60,
"column": 87
} | {
"line": 60,
"column": 89
} | {
"line": 61,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\ninst✝ : SFinite μ\ns : Set Ω\nht : Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ (μ.tilted fun x ↦ t * X x) s = ∫⁻ (a : Ω) in s, ENNReal.ofReal (rexp (t * X a - cgf X μ t)) ∂μ",
"ppTerm": "?m.39",
"assigned": true,
"usedConst... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 66,
"column": 86
} | {
"line": 66,
"column": 88
} | {
"line": 67,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\ns : Set Ω\nhs : MeasurableSet s\n⊢ (μ.tilted fun x ↦ t * X x) s = ENNReal.ofReal (∫ (a : Ω) in s, rexp (t * X a) / mgf X μ t ∂μ)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProdu... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 820,
"column": 78
} | {
"line": 820,
"column": 80
} | {
"line": 821,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 70,
"column": 86
} | {
"line": 70,
"column": 88
} | {
"line": 71,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\ninst✝ : SFinite μ\ns : Set Ω\n⊢ (μ.tilted fun x ↦ t * X x) s = ENNReal.ofReal (∫ (a : Ω) in s, rexp (t * X a) / mgf X μ t ∂μ)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductS... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 204,
"column": 49
} | {
"line": 204,
"column": 51
} | {
"line": 205,
"column": 6
} | [
{
"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... | [] | by | [anonymous] | by |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.