module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 632,
"column": 44
} | {
"line": 632,
"column": 55
} | {
"line": 632,
"column": 56
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, StronglyMeasurable f\n⊢ ∀ f ∈ l, StronglyMeasurable f",
"ppTerm": "?m.35",
"a... | [
"α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, StronglyMeasurable f\n⊢ ∀ f ∈ l, StronglyMeasurable f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 45
} | {
"line": 638,
"column": 46
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\ns : Multiset (α → M)\nhs : ∀ f ∈ s, StronglyMeasurable f\n⊢ StronglyMeasurable fun x ↦ (Multiset.map (fun f ↦ f x) s).prod",
"ppTerm": "?m.20",
"assigned": true,
"... | [
"α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\ns : Multiset (α → M)\nhs : ∀ f ∈ s, StronglyMeasurable f\n⊢ StronglyMeasurable fun x ↦ s.prod x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 648,
"column": 2
} | {
"line": 648,
"column": 40
} | {
"line": 648,
"column": 41
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nι : Type u_6\nf : ι → α → M\ns : Finset ι\nhf : ∀ i ∈ s, StronglyMeasurable (f i)\n⊢ StronglyMeasurable fun a ↦ ∏ i ∈ s, f i a",
"ppTerm": "?m.22",
"assigned": true,
... | [
"α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nι : Type u_6\nf : ι → α → M\ns : Finset ι\nhf : ∀ i ∈ s, StronglyMeasurable (f i)\n⊢ StronglyMeasurable fun a ↦ (∏ c ∈ s, f c) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 188,
"column": 68
} | {
"line": 189,
"column": 37
} | {
"line": 190,
"column": 4
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure ν\nthis✝ : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))\nf : ℕ × ℕ → Set α :=\n fun p ↦\n (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1\ne... | [] | by
rw [this, Set.compl_union_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 218,
"column": 40
} | {
"line": 221,
"column": 28
} | {
"line": 222,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nhs : MeasurableSet s\nhs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ\nhνs : ν s ≠ 0\nthis : ¬SigmaFinite (μ.restrict s)\n⊢ μ s = ∞",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Measure... | [] | by
by_contra h
have h_lt_top : Fact (μ s < ∞) := ⟨Ne.lt_top h⟩
exact this inferInstance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Perfect | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 13
} | {
"line": 133,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PerfectSpace α\nU : Set α\nhU : IsOpen[inst✝¹] U\n⊢ Preperfect U",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PerfectSpace α\nU : Set α\nhU : IsOpen[inst✝¹] U\n⊢ Preperfect U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 29
} | {
"line": 331,
"column": 30
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g μ\n⊢ ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g μ\n⊢ ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 806,
"column": 4
} | {
"line": 807,
"column": 15
} | {
"line": 807,
"column": 16
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nf g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n ↦ ((fun n ↦ SimpleFunc.piecewise s hs (hf.ap... | [
"case pos\nα : Type u_1\nβ : Type u_2\nf g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 (f x))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 808,
"column": 4
} | {
"line": 809,
"column": 15
} | {
"line": 809,
"column": 16
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nf g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nx : α\nhx : x ∉ s\n⊢ Tendsto (fun n ↦ ((fun n ↦ SimpleFunc.piecewise s hs (hf.ap... | [
"case neg\nα : Type u_1\nβ : Type u_2\nf g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nx : α\nhx : x ∉ s\n⊢ Tendsto (fun n ↦ (hg.approx n) x) atTop (𝓝 (g x))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 827,
"column": 4
} | {
"line": 827,
"column": 20
} | {
"line": 827,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : (x : α) → Decidable (x ∈ s)\nf : ↑s → β\nhf : StronglyMeasurable f\ng : ↑sᶜ → β\nhg : StronglyMeasurable g\nhs : MeasurableSet s\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n ↦ ((fun n ↦ SimpleFunc.dite ... | [
"case pos\nα : Type u_1\nβ : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : (x : α) → Decidable (x ∈ s)\nf : ↑s → β\nhf : StronglyMeasurable f\ng : ↑sᶜ → β\nhg : StronglyMeasurable g\nhs : MeasurableSet s\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n ↦ (hf.approx n) ⟨x, ⋯⟩) atTop (𝓝 (f ⟨x,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 828,
"column": 4
} | {
"line": 828,
"column": 20
} | {
"line": 828,
"column": 21
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : (x : α) → Decidable (x ∈ s)\nf : ↑s → β\nhf : StronglyMeasurable f\ng : ↑sᶜ → β\nhg : StronglyMeasurable g\nhs : MeasurableSet s\nx : α\nhx : x ∉ s\n⊢ Tendsto (fun n ↦ ((fun n ↦ SimpleFunc.dite ... | [
"case neg\nα : Type u_1\nβ : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : (x : α) → Decidable (x ∈ s)\nf : ↑s → β\nhf : StronglyMeasurable f\ng : ↑sᶜ → β\nhg : StronglyMeasurable g\nhs : MeasurableSet s\nx : α\nhx : x ∉ s\n⊢ Tendsto (fun n ↦ (hg.approx n) ⟨x, ⋯⟩) atTop (𝓝 (g ⟨x,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 837,
"column": 35
} | {
"line": 837,
"column": 46
} | {
"line": 837,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : MeasurableSingletonClass α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nh : SecondCountableTopologyEither α β\nf : α → β\ns : Set α\nhf : ContinuousOn f s\nhs... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : MeasurableSingletonClass α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nh : SecondCountableTopologyEither α β\nf : α → β\ns : Set α\nhf : ContinuousOn f s\nhs : sᶜ.Counta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 183,
"column": 14
} | {
"line": 183,
"column": 30
} | {
"line": 183,
"column": 31
} | [
{
"pp": "case zero\nα : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhF_meas : AEMeasurable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F... | [
"case zero\nα : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhF_meas : AEMeasurable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a\nh_int_fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 866,
"column": 4
} | {
"line": 866,
"column": 84
} | {
"line": 867,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : α → γ\ng' : γ → β\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\ninst✝ : TopologicalSpace β\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\ny : α\n⊢ Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).extend ... | [
"case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : α → γ\ng' : γ → β\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\ninst✝ : TopologicalSpace β\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\ny : α\n⊢ Tendsto (fun n ↦ (hf.approx n) y) atTop (𝓝 (f y))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 868,
"column": 4
} | {
"line": 868,
"column": 81
} | {
"line": 869,
"column": 6
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : α → γ\ng' : γ → β\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\ninst✝ : TopologicalSpace β\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : γ\nhx : ¬∃ y, g y = x\n⊢ Tendsto (fun n ↦ ((fun n ↦ (... | [
"case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : α → γ\ng' : γ → β\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\ninst✝ : TopologicalSpace β\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : γ\nhx : ¬∃ y, g y = x\n⊢ Tendsto (fun n ↦ (hg'.approx n) x) atTop... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 928,
"column": 2
} | {
"line": 928,
"column": 47
} | {
"line": 929,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : AddZeroClass β\ninst✝ : TopologicalSpace β\nP : (f : α → β) → StronglyMeasurable f → Prop\nind : ∀ (c : β) ⦃s : Set α⦄ (hs : MeasurableSet s), P (s.indicator fun x ↦ c) ⋯\nadd :\n ∀ ⦃f g : α → β⦄ (hf : StronglyMeasurable f) (hg : Strongl... | [] | induction s n using SimpleFunc.induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 429,
"column": 2
} | {
"line": 429,
"column": 44
} | {
"line": 429,
"column": 45
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhr : r ≠ ∞\nh : ¬r = 0\nrinv : r * r⁻¹ = 1\nrinv' : r⁻¹ * r = 1\nthis : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), 1 * f a ∂μ\n⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ",
"ppTerm": "?m.85",
"assigned": true,
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhr : r ≠ ∞\nh : ¬r = 0\nrinv : r * r⁻¹ = 1\nrinv' : r⁻¹ * r = 1\nthis : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), 1 * f a ∂μ\n⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1046,
"column": 6
} | {
"line": 1046,
"column": 51
} | {
"line": 1046,
"column": 52
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ x ∉ s, f x = 0\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).restrict s) n) x) atTop (𝓝 (f x))",... | [
"case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ x ∉ s, f x = 0\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 (f x))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1047,
"column": 6
} | {
"line": 1047,
"column": 65
} | {
"line": 1047,
"column": 66
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ x ∉ s, f x = 0\nx : α\nhx : x ∉ s\n⊢ Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).restrict s) n) x) atTop (𝓝 (f x))",... | [
"case neg\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ x ∉ s, f x = 0\nx : α\nhx : x ∉ s\n⊢ Tendsto (fun n ↦ 0) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 472,
"column": 8
} | {
"line": 472,
"column": 74
} | {
"line": 472,
"column": 74
} | [
{
"pp": "case refine_3\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf✝ : α → ℝ≥0∞\nhf✝ : Measurable f✝\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nhf_mono : Monotone f\nhf_prop : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ.trim hm = ∫⁻ (a : α), f n a ∂μ\n⊢ ⨆ n, ∫⁻ (a : α), f n a ∂μ.trim hm ... | [
"case refine_3\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf✝ : α → ℝ≥0∞\nhf✝ : Measurable f✝\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nhf_mono : Monotone f\nhf_prop : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ.trim hm = ∫⁻ (a : α), f n a ∂μ\n⊢ ⨆ n, ∫⁻ (a : α), f n a ∂μ.trim hm = ⨆ n, ∫⁻ (a... | lintegral_iSup (fun n => Measurable.mono (hf n) hm le_rfl) hf_mono | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 19
} | {
"line": 488,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : AEMeasurable f (μ.trim hm)\ns : Set α\nhs : MeasurableSet s\n⊢ AEMeasurable f ((μ.trim hm).restrict s)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.trim",
"... | [] | exact hf.restrict | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1136,
"column": 6
} | {
"line": 1136,
"column": 21
} | {
"line": 1136,
"column": 22
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞\nh_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n ↦ sup... | [
"α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞\nh_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n ↦ support ⇑(fs n)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.FunctionSeries | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 13
} | {
"line": 111,
"column": 14
} | [
{
"pp": "β : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nι : Type u_4\nf : ι → β → F\nu : ι → ℝ\nhu : Summable u\nhfu : ∀ᶠ (n : ι) in cofinite, ∀ (x : β), ‖f n x‖ ≤ u n\n⊢ ∀ᶠ (n : ι) in cofinite, ∀ x ∈ univ, ‖f n x‖ ≤ u n",
"ppTerm": "?m.52",
"assigned": true,
"us... | [
"β : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nι : Type u_4\nf : ι → β → F\nu : ι → ℝ\nhu : Summable u\nhfu : ∀ᶠ (n : ι) in cofinite, ∀ (x : β), ‖f n x‖ ≤ u n\n⊢ {x | ∃ x_1, u x < ‖f x x_1‖}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 51
} | {
"line": 233,
"column": 2
} | [
{
"pp": "α : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤... | [] | exact ge_of_tendsto' h_tendsto (fun m ↦ h_le _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 46
} | {
"line": 145,
"column": 47
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε\nx y : Y\nz : X\n⊢ glueDist Φ Ψ ε (Sum.inr x) (Sum.inl z) ≤\n glueDist Φ Ψ ε (Sum.inr x) (Sum.inr y) + g... | [
"X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε\nx y : Y\nz : X\n⊢ glueDist Φ Ψ ε (Sum.inl z) (Sum.inr x) ≤\n glueDist Φ Ψ ε (Sum.inl z) (Sum.inr y) + glueDist Φ Ψ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 147,
"column": 4
} | {
"line": 148,
"column": 11
} | {
"line": 148,
"column": 12
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε\nx y : X\nz : Y\n⊢ glueDist Φ Ψ ε (Sum.inl x) (Sum.inr z) ≤\n glueDist Φ Ψ ε (Sum.inl x) (Sum.inl y) + g... | [
"X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε\nx y : X\nz : Y\n⊢ glueDist Ψ Φ ε (Sum.inl z) (Sum.inr x) ≤\n glueDist Ψ Φ ε (Sum.inl z) (Sum.inr y) + glueDist Ψ Φ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UnitInterval | {
"line": 173,
"column": 57
} | {
"line": 173,
"column": 68
} | {
"line": 173,
"column": 69
} | [
{
"pp": "x : ↑I\n⊢ 0 ≤ 1 - ↑x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"HSub.hSub",
"Membership.mem",
"id",
"Real.inst... | [
"x : ↑I\n⊢ ↑x ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UnitInterval | {
"line": 178,
"column": 57
} | {
"line": 178,
"column": 68
} | {
"line": 178,
"column": 69
} | [
{
"pp": "x : ↑I\n⊢ 1 - ↑x ≤ 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Real.instAddMonoid",
"instIsLeftCancelAddOfAddLeftReflectLE",
"Real.instSub",
"covariant_swap_add_of_covarian... | [
"x : ↑I\n⊢ 0 ≤ ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UnitInterval | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 13
} | {
"line": 200,
"column": 14
} | [
{
"pp": "i j : ↑I\nh : 0 < ↑i ∧ ↑j < 1\n⊢ ↑j * ↑i < ↑i",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"i j : ↑I\nh : 0 < ↑i ∧ ↑j < 1\n⊢ ↑j * ↑i < ↑i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 25
} | {
"line": 134,
"column": 4
} | [
{
"pp": "case mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhy : y ∈ cylinder x n\n⊢ cylinder y n = cylinder x n",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"PiNat.cylinder",
"Nat"
],
"usedFVars": [
"E",
"y",
"n",
... | [
"case mp.h₁\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhy : y ∈ cylinder x n\n⊢ cylinder y n ⊆ cylinder x n",
"case mp.h₂\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhy : y ∈ cylinder x n\n⊢ cylinder x n ⊆ cylinder y n"
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 23
} | {
"line": 171,
"column": 24
} | [
{
"pp": "case mp\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nk : E n\nhk : ∀ i < n + 1, y i = update x n k i\ni : ℕ\nhi : i < n\n⊢ y i = x i",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nk : E n\nhk : ∀ i < n + 1, y i = update x n k i\ni : ℕ\nhi : i < n\n⊢ y i = x i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 361,
"column": 6
} | {
"line": 361,
"column": 17
} | {
"line": 361,
"column": 18
} | [
{
"pp": "case inl.inl\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx z y : E i\n⊢ dist ⟨i, x⟩ ⟨i, z⟩ ≤ dist ⟨i, x⟩ ⟨i, y⟩ + dist ⟨i, y⟩ ⟨i, z⟩",
"ppTerm": "?inl.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrA... | [
"case inl.inl\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx z y : E i\n⊢ dist x z ≤ dist x y + dist y z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 365,
"column": 10
} | {
"line": 365,
"column": 47
} | {
"line": 365,
"column": 48
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx : E i\nj : ι\ny : E j\nz : E i\nhij : i ≠ j\n⊢ dist x z ≤ dist x ⋯.some + 0 + 0 + (0 + 0 + dist ⋯.some z)",
"ppTerm": "?m.140",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real... | [
"ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx : E i\nj : ι\ny : E j\nz : E i\nhij : i ≠ j\n⊢ dist x z ≤ dist x ⋯.some + dist ⋯.some z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 83
} | {
"line": 318,
"column": 84
} | [
{
"pp": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : dist x y < (1 / 2) ^ n\ni : ℕ\nhi : i ≤ n\nhne : x ≠ y\n⊢ n < firstDiff x y",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : dist x y < (1 / 2) ^ n\ni : ℕ\nhi : i ≤ n\nhne : x ≠ y\n⊢ n < firstDiff x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 429,
"column": 4
} | {
"line": 433,
"column": 45
} | {
"line": 435,
"column": 0
} | [
{
"pp": "case refine_3.inr\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx : E i\nj : ι\ny : E j\nhij : i ≠ j\n⊢ Sigma.dist ⟨i, x⟩ ⟨j, y⟩ = 0 → ⟨i, x⟩ = ⟨j, y⟩",
"ppTerm": "?refine_3.inr",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"T... | [] | · intro h
apply (lt_irrefl (1 : ℝ) _).elim
calc
1 ≤ Sigma.dist (⟨i, x⟩ : Σ k, E k) ⟨j, y⟩ := Sigma.one_le_dist_of_ne hij _ _
_ < 1 := by rw [h]; exact zero_lt_one | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 450,
"column": 4
} | {
"line": 451,
"column": 60
} | {
"line": 452,
"column": 2
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝¹ : (i : ι) → MetricSpace (E i)\ninst✝ : ∀ (i : ι), CompleteSpace (E i)\ns : ι → Set ((i : ι) × E i) := fun i ↦ Sigma.fst ⁻¹' {i}\nU : Set (((k : ι) × E k) × (k : ι) × E k) := {p | dist p.1 p.2 < 1}\ni : ι\n⊢ IsComplete (s i)",
"ppTerm": "?m.45",
"assigned":... | [] | simp only [s, ← range_sigmaMk]
exact (isometry_mk i).isUniformInducing.isComplete_range | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 450,
"column": 4
} | {
"line": 451,
"column": 60
} | {
"line": 452,
"column": 2
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝¹ : (i : ι) → MetricSpace (E i)\ninst✝ : ∀ (i : ι), CompleteSpace (E i)\ns : ι → Set ((i : ι) × E i) := fun i ↦ Sigma.fst ⁻¹' {i}\nU : Set (((k : ι) × E k) × (k : ι) × E k) := {p | dist p.1 p.2 < 1}\ni : ι\n⊢ IsComplete (s i)",
"ppTerm": "?m.45",
"assigned":... | [] | simp only [s, ← range_sigmaMk]
exact (isometry_mk i).isUniformInducing.isComplete_range | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 333,
"column": 2
} | {
"line": 339,
"column": 36
} | {
"line": 341,
"column": 0
} | [
{
"pp": "case mpr\nE : ℕ → Type u_1\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nf : ((n : ℕ) → E n) → α\n⊢ (∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n) →\n ∀ (x y : (n : ℕ) → E n), dist (f x) (f y) ≤ dist x y",
"ppTerm": "?mpr",
"assigned": true,
"usedConst... | [] | · intro H x y
rcases eq_or_ne x y with (rfl | hne)
· simp [PiNat.dist_nonneg]
rw [dist_eq_of_ne hne]
apply H x y (firstDiff x y)
rw [firstDiff_comm]
exact mem_cylinder_firstDiff _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Polish | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 13
} | {
"line": 294,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsOpen[inst✝¹] s\n⊢ IsClopenable s",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsOpen[inst✝¹] s\n⊢ IsClopenable s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 564,
"column": 35
} | {
"line": 564,
"column": 50
} | {
"line": 564,
"column": 51
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx y : (n : ℕ) × X n\nm : ℕ\nhx : x.fst ≤ m + 1\nhy : y.fst ≤ m + 1\nh : ¬max x.fst y.fst = m + 1\nthis : max x.fst y.fst ≤ m.succ\n⊢ max x.fst y.fst ≤ m",
"ppTerm": "?m.211",
"assi... | [
"X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx y : (n : ℕ) × X n\nm : ℕ\nhx : x.fst ≤ m + 1\nhy : y.fst ≤ m + 1\nh : ¬max x.fst y.fst = m + 1\nthis : max x.fst y.fst ≤ m.succ\n⊢ x.fst ≤ m ∧ y.fst ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 574,
"column": 20
} | {
"line": 574,
"column": 45
} | {
"line": 575,
"column": 2
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx : (n : ℕ) × X n\n⊢ inductiveLimitDist f x x = 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Sigma.... | [] | simp [inductiveLimitDist] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 574,
"column": 20
} | {
"line": 574,
"column": 45
} | {
"line": 575,
"column": 2
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx : (n : ℕ) × X n\n⊢ inductiveLimitDist f x x = 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Sigma.... | [] | simp [inductiveLimitDist] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 574,
"column": 20
} | {
"line": 574,
"column": 45
} | {
"line": 575,
"column": 2
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx : (n : ℕ) × X n\n⊢ inductiveLimitDist f x x = 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Sigma.... | [] | simp [inductiveLimitDist] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 36
} | {
"line": 385,
"column": 4
} | [
{
"pp": "case mpr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nh : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (n : ℕ) → E n), dist x y < ε → y ∈ s\nx : (n : ℕ) → E n\nhx : x ∈ s\n⊢ ∃ t ∈ {s | ∃ x n, s = cylinder x n}, x ∈ t ∧ t ⊆ s",
"ppTerm... | [
"case mpr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nh : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (n : ℕ) → E n), dist x y < ε → y ∈ s\nx : (n : ℕ) → E n\nhx : x ∈ s\nε : ℝ\nεpos : ε > 0\nhε : ∀ (y : (n : ℕ) → E n), dist x y < ε → y ∈ s\n⊢ ∃ t ∈... | rcases h x hx with ⟨ε, εpos, hε⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Dynamics.Ergodic.MeasurePreserving | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 57
} | {
"line": 89,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf : α → β\nhf : MeasurePreserving f μa μb\nh₂ : MeasurableEmbedding f\ns : Set α\n⊢ MeasurePreserving f (μa.restrict s) (μb.restrict (f '' s))",
"ppTerm": "?m.23",
"assigned": fals... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf : α → β\nhf : MeasurePreserving f μa μb\nh₂ : MeasurableEmbedding f\ns : Set α\n⊢ MeasurePreserving f (μa.restrict s) (μb.restrict (f '' s))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.MeasurePreserving | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 15
} | {
"line": 222,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → α\ns : Set α\nhf : MeasurePreserving f μ μ\nhs : NullMeasurableSet s μ\nn : ℕ\nhvol : μ univ < ↑n * μ s\nA : ∀ (m : ℕ), NullMeasurableSet (f^[m] ⁻¹' s) μ\nB : ∀ (m : ℕ), μ (f^[m] ⁻¹' s) = μ s\nthis : μ univ < ∑ m ∈ Finset.range n, μ (f^[m]... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → α\ns : Set α\nhf : MeasurePreserving f μ μ\nhs : NullMeasurableSet s μ\nn : ℕ\nhvol : μ univ < ↑n * μ s\nA : ∀ (m : ℕ), NullMeasurableSet (f^[m] ⁻¹' s) μ\nB : ∀ (m : ℕ), μ (f^[m] ⁻¹' s) = μ s\nthis : μ univ < ∑ m ∈ Finset.range n, μ (f^[m] ⁻¹' s)\n⊢ ∃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 531,
"column": 4
} | {
"line": 531,
"column": 58
} | {
"line": 531,
"column": 59
} | [
{
"pp": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∉ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\n⊢ ∃ y ∈ s, x ∈ cyl... | [
"E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∉ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\n⊢ ∃ y ∈ s, y ∈ cylinder x (Nat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 38
} | {
"line": 129,
"column": 39
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C ... | [
"case refine_3\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C : Set α} → P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 48
} | {
"line": 97,
"column": 49
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : MeasurableSet s\n⊢ μ.real s + μ.real sᶜ = 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real",
"Compl.compl",
"MeasureTheory.Measure.real",
"id",
... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : MeasurableSet s\n⊢ (μ s).toReal + (μ sᶜ).toReal = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 52
} | {
"line": 223,
"column": 53
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : α → Prop\nf✝ : β → α\nf : α → β\nhf : AEMeasurable f μ\n⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)",
"ppTerm": "?pos✝",
"assigned": t... | [
"case pos\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : α → Prop\nf✝ : β → α\nf : α → β\nhf : AEMeasurable f μ\n⊢ μ = 0 ∨ μ univ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 593,
"column": 4
} | {
"line": 593,
"column": 25
} | {
"line": 594,
"column": 4
} | [
{
"pp": "case refine_1\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\n⊢ ra... | [
"case refine_1.h₁\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := ⋯\nfs : ∀ x ∈ s, f x = x\n⊢ range f ⊆ s",
"case refine_1.h₂\nE : ℕ → Ty... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 25
} | {
"line": 263,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : Countable ι\ninst✝ : T2Space α\ns : ι → Set α\nhs : ∀ (n : ι), AnalyticSet (s n)\ni₀ : ι\nβ : ι → Type\nhβ : (n : ι) → TopologicalSpace (β n)\nh'β : ∀ (n : ι), PolishSpace (β n)\nf : (n : ι) → β n → α\nf_cont : ∀ (n : ι), Continuous[hβ n... | [
"case h₁\nα : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : Countable ι\ninst✝ : T2Space α\ns : ι → Set α\nhs : ∀ (n : ι), AnalyticSet (s n)\ni₀ : ι\nβ : ι → Type\nhβ : (n : ι) → TopologicalSpace (β n)\nh'β : ∀ (n : ι), PolishSpace (β n)\nf : (n : ι) → β n → α\nf_cont : ∀ (n : ι), Continuous[hβ n, i... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 46
} | {
"line": 239,
"column": 47
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : ℝ≥0∞\nhμs : p ≤ μ s\ns_mble : MeasurableSet s\nh : IsProbabilityMeasure μ\n⊢ μ sᶜ ≤ 1 - p",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNRe... | [
"case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : ℝ≥0∞\nhμs : p ≤ μ s\ns_mble : MeasurableSet s\nh : IsProbabilityMeasure μ\n⊢ 1 ≤ 1 - p + μ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 189,
"column": 6
} | {
"line": 191,
"column": 22
} | {
"line": 192,
"column": 4
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\nh : Disjoint μ ν\nε : ℝ≥0\nhε : 0 < ε\nh₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0\nn : ℕ\n⊢ ∃ x ∈ {m | ∃ t, m = μ t + ν tᶜ}, x < ↑ε * (1 / 2) ^ n",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSem... | [] | refine exists_lt_of_csInf_lt ⟨ν univ, ∅, by simp⟩ <| h₁ ▸ ENNReal.mul_pos ?_ (by simp)
norm_cast
exact hε.ne.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 189,
"column": 6
} | {
"line": 191,
"column": 22
} | {
"line": 192,
"column": 4
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\nh : Disjoint μ ν\nε : ℝ≥0\nhε : 0 < ε\nh₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0\nn : ℕ\n⊢ ∃ x ∈ {m | ∃ t, m = μ t + ν tᶜ}, x < ↑ε * (1 / 2) ^ n",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSem... | [] | refine exists_lt_of_csInf_lt ⟨ν univ, ∅, by simp⟩ <| h₁ ▸ ENNReal.mul_pos ?_ (by simp)
norm_cast
exact hε.ne.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 83,
"column": 13
} | {
"line": 83,
"column": 28
} | {
"line": 83,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\na : α\nh : dirac a = 0\n⊢ False",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : MeasurableSpace α\na : α\nh : dirac a = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 271,
"column": 8
} | {
"line": 271,
"column": 35
} | {
"line": 272,
"column": 8
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : Countable ι\ninst✝ : T2Space α\ns : ι → Set α\nhs : ∀ (n : ι), AnalyticSet (s n)\ni₀ : ι\nβ : ι → Type\nhβ : (n : ι) → TopologicalSpace (β n)\nh'β : ∀ (n : ι), PolishSpace (β n)\nf : (n : ι) → β n → α\nf_cont : ∀ (n : ι), Continuous[hβ n... | [
"α : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : Countable ι\ninst✝ : T2Space α\ns : ι → Set α\nhs : ∀ (n : ι), AnalyticSet (s n)\ni₀ : ι\nβ : ι → Type\nhβ : (n : ι) → TopologicalSpace (β n)\nh'β : ∀ (n : ι), PolishSpace (β n)\nf : (n : ι) → β n → α\nf_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f... | rw [← mem_range, f_range n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 139,
"column": 45
} | {
"line": 139,
"column": 56
} | {
"line": 139,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Countable α\ninst✝ : MeasurableSingletonClass α\nμ : Measure α\n⊢ (sum fun a ↦ μ {a} • dirac a) = μ",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Countable α\ninst✝ : MeasurableSingletonClass α\nμ : Measure α\n⊢ (sum fun a ↦ μ {a} • dirac a) = μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 58,
"column": 48
} | {
"line": 58,
"column": 86
} | {
"line": 58,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : Set α\ns_fin : s.Finite\ns_mble : MeasurableSet s\n⊢ MeasurableSet ↑s_fin.toFinset",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Finset",
"id",
"Set.Finite.co... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : Set α\ns_fin : s.Finite\ns_mble : MeasurableSet s\n⊢ MeasurableSet s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 20
} | {
"line": 104,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : Set α\nh : count s = 0\nx : α\nhx : x ∈ s\n⊢ False",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : Set α\nh : count s = 0\nx : α\nhx : x ∈ s\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 39
} | {
"line": 144,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β → α\nhf : Function.Injective f\ns : Finset β\ns_mble : MeasurableSet ↑s\nfs_mble : MeasurableSet (f '' ↑s)\n⊢ MeasurableSet ↑(Finset.image f s)",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β → α\nhf : Function.Injective f\ns : Finset β\ns_mble : MeasurableSet ↑s\nfs_mble : MeasurableSet (f '' ↑s)\n⊢ MeasurableSet (f '' ↑s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 170,
"column": 22
} | {
"line": 170,
"column": 33
} | {
"line": 170,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ns : Set α\ninst✝ : Nonempty α\nh : count = 0\n⊢ False",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ns : Set α\ninst✝ : Nonempty α\nh : count = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 249,
"column": 51
} | {
"line": 249,
"column": 62
} | {
"line": 249,
"column": 63
} | [
{
"pp": "δ : Type u_3\nι : Type u_4\nmδ : MeasurableSpace δ\nc : ι → ℝ\nd : ι → δ\nh1 : ∀ (i : ι), 0 ≤ c i\nh2 : HasSum c 1\n⊢ HasSum (fun i ↦ (c i).toNNReal) 1",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"δ : Type u_3\nι : Type u_4\nmδ : MeasurableSpace δ\nc : ι → ℝ\nd : ι → δ\nh1 : ∀ (i : ι), 0 ≤ c i\nh2 : HasSum c 1\n⊢ HasSum (fun i ↦ (c i).toNNReal) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 42
} | {
"line": 37,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ ∫⁻ (x : α), c ∂μ < ∞",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ ∫⁻ (x : α), c ∂μ < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 13
} | {
"line": 57,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → ℝ≥0∞\nf_bdd : ∃ c, ∀ (x : α), f x ≤ ↑c\n⊢ ∃ y, ∀ x ∈ univ, f x ≤ ↑y",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"congrArg",
"Set.me... | [
"α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → ℝ≥0∞\nf_bdd : ∃ c, ∀ (x : α), f x ≤ ↑c\n⊢ ∃ y, ∀ (x : α), f x ≤ ↑y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 297,
"column": 6
} | {
"line": 298,
"column": 49
} | {
"line": 298,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : dirac x = dirac y\nA : Set α\nA_mble : MeasurableSet A\nobs : A.indicator 1 x = A.indicator 1 y\nx_in_A : x ∈ A\n⊢ x ∈ A ↔ y ∈ A",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : dirac x = dirac y\nA : Set α\nA_mble : MeasurableSet A\nobs : A.indicator 1 x = A.indicator 1 y\nx_in_A : x ∈ A\n⊢ y ∈ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 299,
"column": 6
} | {
"line": 300,
"column": 91
} | {
"line": 300,
"column": 92
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : dirac x = dirac y\nA : Set α\nA_mble : MeasurableSet A\nobs : A.indicator 1 x = A.indicator 1 y\nx_in_A : x ∉ A\n⊢ x ∈ A ↔ y ∈ A",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq... | [
"case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : dirac x = dirac y\nA : Set α\nA_mble : MeasurableSet A\nobs : A.indicator 1 x = A.indicator 1 y\nx_in_A : x ∉ A\n⊢ y ∉ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 656,
"column": 14
} | {
"line": 656,
"column": 90
} | {
"line": 657,
"column": 10
} | [
{
"pp": "case inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n ... | [] | exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 27
} | {
"line": 193,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nμ : Measure α\ninst✝ : SFinite μ\nh : IsFiniteMeasure μ\nn : ℕ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ g ≤ ↑n ∧ ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g a ∂μ",
"ppTerm": "?m.230",
"assigned": false,
"usedConstants": [],
"... | [
"α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nμ : Measure α\ninst✝ : SFinite μ\nh : IsFiniteMeasure μ\nn : ℕ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ g ≤ ↑n ∧ ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g a ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 61,
"column": 96
} | {
"line": 66,
"column": 57
} | {
"line": 70,
"column": 0
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α✝\nmβ : MeasurableSpace β\nα : Type u_3\nm : MeasurableSpace α\n⊢ MeasurableAdd₂ (Measure α)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"MeasureTheory.Measure.instMeasurableSpace",
"Measure... | [] | by
refine ⟨Measure.measurable_of_measurable_coe _ fun s hs => ?_⟩
simp_rw [Measure.coe_add, Pi.add_apply]
refine Measurable.add ?_ ?_
· exact (Measure.measurable_coe hs).comp measurable_fst
· exact (Measure.measurable_coe hs).comp measurable_snd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 45
} | {
"line": 94,
"column": 46
} | [
{
"pp": "case iUnion\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : α → Measure β\ninst✝ : ∀ (a : α), IsFiniteMeasure (μ a)\nS : Set (Set β)\nhgen : mβ = MeasurableSpace.generateFrom S\nhpi : IsPiSystem S\nh_basic : ∀ s ∈ S, Measurable fun a ↦ (μ a) s\nh_univ : Measurable fun a... | [
"case iUnion\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : α → Measure β\ninst✝ : ∀ (a : α), IsFiniteMeasure (μ a)\nS : Set (Set β)\nhgen : mβ = MeasurableSpace.generateFrom S\nhpi : IsPiSystem S\nh_basic : ∀ s ∈ S, Measurable fun a ↦ (μ a) s\nh_univ : Measurable fun a ↦ (μ a) uni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 87
} | {
"line": 243,
"column": 88
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ∞\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas :... | [
"α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ∞\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 74
} | {
"line": 58,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nm : MeasurableSpace X\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nhU : IsOpen[inst✝¹] U\n⊢ μ U = 0 ↔ U = ∅",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nm : MeasurableSpace X\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nhU : IsOpen[inst✝¹] U\n⊢ μ U = 0 ↔ U = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 44
} | {
"line": 119,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\nm : MeasurableSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nf g : X → Y\nhU : IsOpen[inst✝³] U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : ∀ᵐ (x : X) ∂μ, x ∈ U → f x = g x\n... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\nm : MeasurableSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nf g : X → Y\nhU : IsOpen[inst✝³] U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : μ {a | a ∈ U ∧ ¬f a = g a} = 0\n⊢ EqOn f g U"
... | simp only [ae_iff, Classical.not_imp] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 779,
"column": 4
} | {
"line": 779,
"column": 40
} | {
"line": 779,
"column": 41
} | [
{
"pp": "α : Type u_2\ninst✝³ : MetricSpace α\ninst✝² : CompleteSpace α\ninst✝¹ : SecondCountableTopology α\ninst✝ : Nonempty α\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : ℕ → α\nhu : DenseRange u\ns : Set (ℕ → ℕ) := ⋯\ng : ↑s → α := ⋯\nA : ∀ (x : ↑s) (n : ℕ), dist (g x... | [
"α : Type u_2\ninst✝³ : MetricSpace α\ninst✝² : CompleteSpace α\ninst✝¹ : SecondCountableTopology α\ninst✝ : Nonempty α\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : ℕ → α\nhu : DenseRange u\ns : Set (ℕ → ℕ) := ⋯\ng : ↑s → α := ⋯\nA : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 41
} | {
"line": 819,
"column": 42
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → EDist (F i)\nx y : (i : ι) → F i\ni : ι\nh : edist x y < 2⁻¹ ^ encode i\n⊢ edist (x i) (y i) ≤ edist x y",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → EDist (F i)\nx y : (i : ι) → F i\ni : ι\nh : edist x y < 2⁻¹ ^ encode i\n⊢ edist (x i) (y i) ≤ edist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 853,
"column": 18
} | {
"line": 853,
"column": 29
} | {
"line": 853,
"column": 30
} | [
{
"pp": "E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ ε / 2 > 0",
"ppTerm": "?m.296",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"instHDiv",
"and... | [
"E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ ¬ε = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 855,
"column": 48
} | {
"line": 855,
"column": 59
} | {
"line": 855,
"column": 60
} | [
{
"pp": "E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\n⊢ ε / 2 ≠ 0",
"ppTerm": "?m.350",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\n⊢ ¬ε = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 873,
"column": 54
} | {
"line": 873,
"column": 65
} | {
"line": 873,
"column": 66
} | [
{
"pp": "E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ * ↑K.card < ε / 2\nx y : (i : ι) → F i\nhxy : ∀ (x_1 : ι) (h : x_... | [
"E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ * ↑K.card < ε / 2\nx y : (i : ι) → F i\nhxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 877,
"column": 42
} | {
"line": 877,
"column": 64
} | {
"line": 877,
"column": 65
} | [
{
"pp": "case h₁\nE : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ * ↑K.card < ε / 2\nx y : (i : ι) → F i\nhxy : ∀ (x_1 : ι... | [
"case h₁\nE : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ * ↑K.card < ε / 2\nx y : (i : ι) → F i\nhxy : ∀ (x_1 : ι) (h : x_1 ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 926,
"column": 4
} | {
"line": 926,
"column": 25
} | {
"line": 926,
"column": 26
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoMetricSpace (F i)\nx y : (i : ι) → F i\n⊢ Summable fun i ↦ 2⁻¹ ^ encode i",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionCommMonoid.toDivisionMonoid",
... | [
"ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoMetricSpace (F i)\nx y : (i : ι) → F i\n⊢ Summable fun i ↦ (2 ^ encode i)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 934,
"column": 2
} | {
"line": 934,
"column": 41
} | {
"line": 934,
"column": 42
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoMetricSpace (F i)\nx y : (i : ι) → F i\ni : ι\nh : dist x y < 2⁻¹ ^ encode i\n⊢ dist (x i) (y i) ≤ dist x y",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoMetricSpace (F i)\nx y : (i : ι) → F i\ni : ι\nh : dist x y < 2⁻¹ ^ encode i\n⊢ dist (x i) (y i) ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Convolution | {
"line": 47,
"column": 61
} | {
"line": 48,
"column": 45
} | {
"line": 50,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nμ ν : Measure M\nf : M → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (z : M), f z ∂μ ∗ₘ ν = ∫⁻ (z : M × M), f (z.1 * z.2) ∂μ.prod ν",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Measu... | [] | by
rw [mconv, lintegral_map hf measurable_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1135,
"column": 6
} | {
"line": 1135,
"column": 95
} | {
"line": 1135,
"column": 96
} | [
{
"pp": "X : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis : Nonempty X\nn : ℕ\nhn :... | [
"X : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis : Nonempty X\nn : ℕ\nhn : dist x (den... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1138,
"column": 4
} | {
"line": 1138,
"column": 15
} | {
"line": 1138,
"column": 16
} | [
{
"pp": "case inr.refine_2\nX : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis✝ : Non... | [
"case inr.refine_2\nX : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis✝ : Nonempty X\nn :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 79,
"column": 14
} | {
"line": 80,
"column": 11
} | {
"line": 80,
"column": 12
} | [
{
"pp": "case basic\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : IsFiniteMeasure ν\ns✝ : Set (α × β)\ns : Set α\nhs : s ∈ {s | MeasurableSet s}\nt : Set β\n⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' (fun x1 x2 ↦ x1 ×ˢ x2) s t)",
"ppTerm": "?basic",
... | [
"case basic\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : IsFiniteMeasure ν\ns✝ : Set (α × β)\ns : Set α\nhs : s ∈ {s | MeasurableSet s}\nt : Set β\n⊢ Measurable fun x ↦ s.indicator (fun x ↦ ν t) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 27
} | {
"line": 88,
"column": 28
} | [
{
"pp": "case iUnion\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : IsFiniteMeasure ν\ns : Set (α × β)\nf : ℕ → Set (α × β)\nhfd : Pairwise (Disjoint on f)\nhfm : ∀ (i : ℕ), MeasurableSet (f i)\nihf : ∀ (i : ℕ), Measurable fun x ↦ ν (Prod.mk x ⁻¹' f i... | [
"case iUnion\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : IsFiniteMeasure ν\ns : Set (α × β)\nf : ℕ → Set (α × β)\nhfd : Pairwise (Disjoint on f)\nhfm : ∀ (i : ℕ), MeasurableSet (f i)\nihf : ∀ (i : ℕ), Measurable fun x ↦ ν (Prod.mk x ⁻¹' f i)\nthis : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 25
} | {
"line": 100,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' s)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheor... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x ↦ (sum (sfiniteSeq ν)) (Prod.mk x ⁻¹' s)"
] | rw [← sum_sfiniteSeq ν] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Complement | {
"line": 145,
"column": 31
} | {
"line": 145,
"column": 52
} | {
"line": 145,
"column": 53
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\nh : IsComplement S T\nx : G\n⊢ x ∈ S * T",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.GroupTheory.Complement.0.Subgroup.IsComplement.mul_eq._simp_1_1",
"HMul.hMul",
"Monoid.toMulOneC... | [
"G : Type u_1\ninst✝ : Group G\nS T : Set G\nh : IsComplement S T\nx : G\n⊢ ∃ x_1 ∈ S, ∃ y ∈ T, x_1 * y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 149,
"column": 13
} | {
"line": 149,
"column": 43
} | {
"line": 149,
"column": 44
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nT : Set G\nh : IsComplement ∅ T\n⊢ False",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nT : Set G\nh : IsComplement ∅ T\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 153,
"column": 13
} | {
"line": 153,
"column": 43
} | {
"line": 153,
"column": 44
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Set G\nh : IsComplement S ∅\n⊢ False",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nS : Set G\nh : IsComplement S ∅\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 38
} | {
"line": 282,
"column": 0
} | [
{
"pp": "case right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ : Measure γ\ninst✝⁵ : SFinite ν✝\nX : Type u_4\nY : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace Y\nm : M... | [] | exact v_open.measure_pos ν ⟨y, yv⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 38
} | {
"line": 282,
"column": 0
} | [
{
"pp": "case right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ : Measure γ\ninst✝⁵ : SFinite ν✝\nX : Type u_4\nY : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace Y\nm : M... | [] | exact v_open.measure_pos ν ⟨y, yv⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 38
} | {
"line": 282,
"column": 0
} | [
{
"pp": "case right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ : Measure γ\ninst✝⁵ : SFinite ν✝\nX : Type u_4\nY : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace Y\nm : M... | [] | exact v_open.measure_pos ν ⟨y, yv⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 45
} | {
"line": 223,
"column": 46
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nH : μ.InnerRegularWRT p q\nhU : q U\nr : ℝ≥0∞\nhr : r < μ U\n⊢ r < ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nH : μ.InnerRegularWRT p q\nhU : q U\nr : ℝ≥0∞\nhr : r < μ U\n⊢ ∃ i ⊆ U, p i ∧ r < μ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 202,
"column": 65
} | {
"line": 202,
"column": 76
} | {
"line": 202,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑S\nhy : (fun s ↦ (↑s)⁻¹ * g ∈ T) y\n⊢ ?m.180 = ?m.181",
"ppTerm": "?m.184",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑S\nhy : (fun s ↦ (↑s)⁻¹ * g ∈ T) y\n⊢ ?m.180 = ?m.181"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 210,
"column": 65
} | {
"line": 210,
"column": 76
} | {
"line": 210,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑T\nhy : (fun t ↦ g * (↑t)⁻¹ ∈ S) y\n⊢ ?m.180 = ?m.181",
"ppTerm": "?m.184",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑T\nhy : (fun t ↦ g * (↑t)⁻¹ ∈ S) y\n⊢ ?m.180 = ?m.181"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 46
} | {
"line": 267,
"column": 46
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\n⊢ IsComplement (↑H) (range f)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\n⊢ Bijective ((range f).restrict Quotient.mk'')"
] | isComplement_subgroup_left_iff_bijective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 55
} | {
"line": 422,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ μ' : Measure α\nν ν' : Measure β\ninst✝ : SFinite ν'\nh1 : μ ≪ μ'\nh2 : ν ≪ ν'\ns : Set (α × β)\nhs : MeasurableSet s\nh2s : (fun x ↦ ν' (Prod.mk x ⁻¹' s)) =ᵐ[μ'] 0\n⊢ (fun x ↦ ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0",
"ppTe... | [] | exact (h2s.filter_mono h1.ae_le).mono fun _ h => h2 h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 45
} | {
"line": 372,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nA : Set α\nμ : Measure α\ninst✝ : μ.OuterRegular\nr : ℝ≥0∞\nhr : μ A < r\n⊢ ⨅ U, ⨅ (_ : A ⊆ U), ⨅ (_ : IsOpen[inst✝¹] U), μ U < r",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTh... | [
"α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nA : Set α\nμ : Measure α\ninst✝ : μ.OuterRegular\nr : ℝ≥0∞\nhr : μ A < r\n⊢ ∃ i, A ⊆ i ∧ IsOpen[inst✝¹] i ∧ μ i < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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