module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 804,
"column": 36
} | {
"line": 804,
"column": 47
} | {
"line": 804,
"column": 48
} | [
{
"pp": "M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, Measurable f\n⊢ ∀ f ∈ l, Measurable f",
"ppTerm": "?m.36",
"assigned": false,... | [
"M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, Measurable f\n⊢ ∀ f ∈ l, Measurable f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 810,
"column": 2
} | {
"line": 810,
"column": 13
} | {
"line": 810,
"column": 14
} | [
{
"pp": "case mk\nM : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEMeasurable f μ\n⊢ AEMeasurable (prod (Quot.mk (⇑(List.isSet... | [
"case mk\nM : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEMeasurable f μ\n⊢ AEMeasurable l.prod μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 810,
"column": 38
} | {
"line": 810,
"column": 49
} | {
"line": 810,
"column": 50
} | [
{
"pp": "M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEMeasurable f μ\n⊢ ∀ f ∈ l, AEMeasurable f μ",
"ppTerm": "?m.37",... | [
"M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEMeasurable f μ\n⊢ ∀ f ∈ l, AEMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1418,
"column": 2
} | {
"line": 1418,
"column": 29
} | {
"line": 1418,
"column": 30
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nι : Type u_8\nι' : Type u_9\ne : ι' ≃ ι\nm : ι → Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ (sum (m ∘ ⇑e)) s = (sum m) s",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Equiv.instEquivLi... | [
"α : Type u_1\nm0 : MeasurableSpace α\nι : Type u_8\nι' : Type u_9\ne : ι' ≃ ι\nm : ι → Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ ∑' (i : ι'), (m (e i)) s = ∑' (i : ι), (m i) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 815,
"column": 2
} | {
"line": 815,
"column": 45
} | {
"line": 815,
"column": 46
} | [
{
"pp": "M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\ns : Multiset (α → M)\nhs : ∀ f ∈ s, Measurable f\n⊢ Measurable fun x ↦ (map (fun f ↦ f x) s).prod",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"... | [
"M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\ns : Multiset (α → M)\nhs : ∀ f ∈ s, Measurable f\n⊢ Measurable fun x ↦ s.prod x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 820,
"column": 2
} | {
"line": 820,
"column": 45
} | {
"line": 820,
"column": 46
} | [
{
"pp": "M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\ns : Multiset (α → M)\nhs : ∀ f ∈ s, AEMeasurable f μ\n⊢ AEMeasurable (fun x ↦ (map (fun f ↦ f x) s).prod) μ",
"ppTerm": "?m.21",
"assigned": true,
"... | [
"M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\ns : Multiset (α → M)\nhs : ∀ f ∈ s, AEMeasurable f μ\n⊢ AEMeasurable (fun x ↦ s.prod x) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 852,
"column": 2
} | {
"line": 852,
"column": 40
} | {
"line": 852,
"column": 41
} | [
{
"pp": "M : Type u_2\nι : Type u_3\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\nf : ι → α → M\ns : Finset ι\nhf : ∀ i ∈ s, AEMeasurable (f i) μ\n⊢ AEMeasurable (fun a ↦ ∏ i ∈ s, f i a) μ",
"ppTerm": "?m.23",
"assigned"... | [
"M : Type u_2\nι : Type u_3\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\nf : ι → α → M\ns : Finset ι\nhf : ∀ i ∈ s, AEMeasurable (f i) μ\n⊢ AEMeasurable (fun a ↦ (∏ c ∈ s, f c) a) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 189,
"column": 3
} | {
"line": 189,
"column": 41
} | {
"line": 189,
"column": 41
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns t u : Set α✝\nα : Type u_6\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nhα : BorelSpace α\np : α → Prop\n⊢ instMeasurableSpace = borel (Subtype p)",
"ppTerm": "?m.9",
"assigned": true,
"usedCon... | [] | by borelize α; symm; apply borel_comap | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 341,
"column": 6
} | {
"line": 341,
"column": 35
} | {
"line": 341,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : OpensMeasurableSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : OpensMeasurableSpace β\ninst✝⁷ : TopologicalS... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : OpensMeasurableSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : OpensMeasurableSpace β\ninst✝⁷ : TopologicalSpace γ\ninst... | inseparable_iff_forall_isOpen | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 69,
"column": 10
} | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\nthis : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)\nH : ∀ (a : α), MeasurableSet (Iio a)\na : α\nhcovBy : ¬∃ b, a ⋖ b\nt : Set α\nhat : t ⊆ Ioi a\nhtc : t.Co... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\nthis : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)\nH : ∀ (a : α), MeasurableSet (Iio a)\na : α\nhcovBy : ¬∃ b, a ⋖ b\nt : Set α\nhat : t ⊆ Ioi a\nhtc : t.Countable\nhtU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 416,
"column": 6
} | {
"line": 416,
"column": 27
} | {
"line": 417,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : OpensMeasurableSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : OpensMeasurableSpace β\ninst✝⁶ : TopologicalS... | [
"case h₁\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : OpensMeasurableSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : OpensMeasurableSpace β\ninst✝⁶ : TopologicalSpac... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 437,
"column": 6
} | {
"line": 437,
"column": 27
} | {
"line": 438,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : OpensMeasurableSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : OpensMeasurableSpace β\ninst✝⁶ : TopologicalS... | [
"case h₁\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : OpensMeasurableSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : OpensMeasurableSpace β\ninst✝⁶ : TopologicalSpac... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 147,
"column": 37
} | {
"line": 147,
"column": 66
} | {
"line": 147,
"column": 67
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) < ∞\n⊢ 0 < ↑ε / 2",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"E... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) < ∞\n⊢ ¬ε = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 157,
"column": 57
} | {
"line": 157,
"column": 75
} | {
"line": 157,
"column": 75
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) < ∞\nε0 : 0 < ↑ε / 2\nthis :\n infEDist x (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) <\n infEDist x (closure[Pse... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) < ∞\nε0 : 0 < ↑ε / 2\nthis :\n infEDist x (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) <\n infEDist x (closure[PseudoEMetricSp... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 217,
"column": 24
} | {
"line": 217,
"column": 35
} | {
"line": 217,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nU : Set α\nhU : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n ↦ (fun x ↦ infEDist x Uᶜ) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\n⊢ x ∉ Uᶜ",
"ppTerm... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nU : Set α\nhU : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n ↦ (fun x ↦ infEDist x Uᶜ) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\n⊢ x ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 219,
"column": 37
} | {
"line": 219,
"column": 66
} | {
"line": 219,
"column": 67
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nU : Set α\nhU : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n ↦ (fun x ↦ infEDist x Uᶜ) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬infEDist x Uᶜ ... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nU : Set α\nhU : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n ↦ (fun x ↦ infEDist x Uᶜ) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬infEDist x Uᶜ = 0\n⊢ ¬infE... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 667,
"column": 55
} | {
"line": 667,
"column": 85
} | {
"line": 667,
"column": 86
} | [
{
"pp": "α : Type u_6\nβ : Type u_7\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : TopologicalSpace β\nhβ : BorelSpace β\ne : α → β\nh'e : MeasurableEmbedding e\nh''e : IsInducing e\n⊢ MeasurableSpace.comap e (borel β) = inst✝³",
"ppTerm": "?m.30",
"assigne... | [
"α : Type u_6\nβ : Type u_7\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : TopologicalSpace β\nhβ : BorelSpace β\ne : α → β\nh'e : MeasurableEmbedding e\nh''e : IsInducing e\n⊢ MeasurableSpace.comap e (borel β) = inst✝³"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 52
} | {
"line": 342,
"column": 4
} | [
{
"pp": "α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l ≤ u ∧ Icc l u = S}",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l ≤ u ∧ Icc l u = S}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 31
} | {
"line": 134,
"column": 32
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\nx : α\nx_in_E : x ∈ E\ny : α\nhy : ENNReal.ofReal δ ≤ infEDist y E\n⊢ ENNReal.ofReal δ ≤ edist x y",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\nx : α\nx_in_E : x ∈ E\ny : α\nhy : ENNReal.ofReal δ ≤ infEDist y E\n⊢ ENNReal.ofReal δ ≤ edist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 32
} | {
"line": 141,
"column": 33
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\n⊢ Eᶜᶜ ⊆ (thickening δ (thickening δ E)ᶜ)ᶜ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"id",
"LE.le",
"Set.instCompl",
... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\n⊢ E ⊆ (thickening δ (thickening δ E)ᶜ)ᶜ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 312,
"column": 39
} | {
"line": 312,
"column": 68
} | {
"line": 312,
"column": 69
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns t : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x s + hausdorffEDist s t < ∞\n⊢ ↑ε / 2 ≠ 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"ENNReal.ofNNReal",
"instHDiv",
"congrArg",... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns t : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x s + hausdorffEDist s t < ∞\n⊢ ¬ε = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 325,
"column": 30
} | {
"line": 325,
"column": 48
} | {
"line": 325,
"column": 48
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns t : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x s + hausdorffEDist s t < ∞\nε0 : ↑ε / 2 ≠ 0\nthis✝ : infEDist x s < infEDist x s + ↑ε / 2\ny : α\nys : y ∈ s\ndxy : edist x y < infEDist x s + ↑ε / 2\nthis : hausdorffEDist s t < hausdorffEDist s t + ↑ε ... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns t : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x s + hausdorffEDist s t < ∞\nε0 : ↑ε / 2 ≠ 0\nthis✝ : infEDist x s < infEDist x s + ↑ε / 2\ny : α\nys : y ∈ s\ndxy : edist x y < infEDist x s + ↑ε / 2\nthis : hausdorffEDist s t < hausdorffEDist s t + ↑ε / 2\nz : α\n... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 349,
"column": 4
} | {
"line": 353,
"column": 100
} | {
"line": 354,
"column": 2
} | [
{
"pp": "case left\nα : Type u\ninst✝ : PseudoEMetricSpace α\ns t u : Set α\n⊢ ∀ x ∈ s, infEDist x u ≤ hausdorffEDist s t + hausdorffEDist t u",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"le_refl",
"Trans.trans",
"ENNReal.instAddCommMonoid",
... | [] | exact fun x xs =>
calc
infEDist x u ≤ infEDist x t + hausdorffEDist t u :=
infEDist_le_infEDist_add_hausdorffEDist
_ ≤ hausdorffEDist s t + hausdorffEDist t u := by grw [infEDist_le_hausdorffEDist_of_mem xs] | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 13
} | {
"line": 402,
"column": 14
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nx : α\nxs : x ∈ s\nthis : infEDist x ∅ ≤ hausdorffEDist s ∅\n⊢ hausdorffEDist s ∅ = ∞",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nx : α\nxs : x ∈ s\nthis : infEDist x ∅ ≤ hausdorffEDist s ∅\n⊢ hausdorffEDist s ∅ = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 357,
"column": 85
} | {
"line": 387,
"column": 61
} | {
"line": 389,
"column": 0
} | [
{
"pp": "α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\n⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l ∈ s, ∃ u ∈ s, l ... | [] | by
set S : Set (Set α) := { S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S }
refine le_antisymm ?_ (generateFrom_Ico_mem_le_borel _ _)
letI : MeasurableSpace α := generateFrom S
rw [borel_eq_generateFrom_Iio]
refine generateFrom_le (forall_mem_range.2 fun a => ?_)
rcases hd.exists_countable_dense_subset_bot_top ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 52
} | {
"line": 400,
"column": 4
} | [
{
"pp": "α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ico l u = S}",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ico l u = S}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 590,
"column": 2
} | {
"line": 591,
"column": 9
} | {
"line": 591,
"column": 10
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\nhs : s.Nonempty\n⊢ IsGLB ((fun x_1 ↦ dist x x_1) '' s) (infDist x s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"iInf",
"congrArg",
"Metric.infDist"... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\nhs : s.Nonempty\n⊢ IsGLB ((fun x_1 ↦ dist x x_1) '' s) (⨅ y, dist x ↑y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 23
} | {
"line": 475,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty\nE : Set α\n⊢ cthickening δ E = ⋂ ε ∈ s, cthickening ε E",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Real",
"Se... | [
"case h₁\nα : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty\nE : Set α\n⊢ cthickening δ E ⊆ ⋂ ε ∈ s, cthickening ε E",
"case h₂\nα : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 425,
"column": 2
} | {
"line": 425,
"column": 52
} | {
"line": 426,
"column": 4
} | [
{
"pp": "α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ioc l u = S}",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ioc l u = S}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 744,
"column": 2
} | {
"line": 744,
"column": 36
} | {
"line": 744,
"column": 37
} | [
{
"pp": "α : Type u\ninst✝¹ : PseudoMetricSpace α\ns : Set α\ninst✝ : ProperSpace α\nhne : s.Nonempty\nx : α\n⊢ ∃ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, infDist x s = dist x y",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"α : Type u\ninst✝¹ : PseudoMetricSpace α\ns : Set α\ninst✝ : ProperSpace α\nhne : s.Nonempty\nx : α\n⊢ ∃ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, infDist x s = dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 522,
"column": 2
} | {
"line": 522,
"column": 23
} | {
"line": 523,
"column": 2
} | [
{
"pp": "case neg\nα : Type u\ninst✝ : PseudoEMetricSpace α\nE : Set α\ns : Set ℝ\nhs : ∀ (ε : ℝ), 0 < ε → (s ∩ Ioc 0 ε).Nonempty\nhs₀ : ¬s ⊆ Ioi 0\nδ : ℝ\nhδs : δ ∈ s\nδ_nonpos : δ ≤ 0\n⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E = ⋂ δ ∈ s, cthickening δ E",
"ppTerm": "?neg✝",
"as... | [
"case neg.h₁\nα : Type u\ninst✝ : PseudoEMetricSpace α\nE : Set α\ns : Set ℝ\nhs : ∀ (ε : ℝ), 0 < ε → (s ∩ Ioc 0 ε).Nonempty\nhs₀ : ¬s ⊆ Ioi 0\nδ : ℝ\nhδs : δ ∈ s\nδ_nonpos : δ ≤ 0\n⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E ⊆ ⋂ δ ∈ s, cthickening δ E",
"case neg.h₂\nα : Type u\ninst✝ : Pse... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 13
} | {
"line": 557,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : PseudoMetricSpace α\nx : α\nE : Set α\nhx : x ∈ E\nδ : ℝ\n⊢ {x} ⊆ E",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Set.instSingletonSet",
"id",
"LE.le",
"Set.instLE",
"Singleton.singleton... | [
"α : Type u_2\ninst✝ : PseudoMetricSpace α\nx : α\nE : Set α\nhx : x ∈ E\nδ : ℝ\n⊢ x ∈ E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 836,
"column": 6
} | {
"line": 836,
"column": 91
} | {
"line": 836,
"column": 92
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\nr : ℝ\nhr : 0 ≤ r\nH1 : ∀ x ∈ s, infDist x t ≤ r\nH2 : ∀ x ∈ t, infDist x s ≤ r\nhs : s.Nonempty\nht : t.Nonempty\n⊢ ∀ x ∈ s, infEDist x t ≤ ENNReal.ofReal r",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\nr : ℝ\nhr : 0 ≤ r\nH1 : ∀ x ∈ s, infDist x t ≤ r\nH2 : ∀ x ∈ t, infDist x s ≤ r\nhs : s.Nonempty\nht : t.Nonempty\n⊢ ∀ x ∈ s, infEDist x t ≤ ENNReal.ofReal r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 837,
"column": 6
} | {
"line": 837,
"column": 91
} | {
"line": 837,
"column": 92
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\nr : ℝ\nhr : 0 ≤ r\nH1 : ∀ x ∈ s, infDist x t ≤ r\nH2 : ∀ x ∈ t, infDist x s ≤ r\nhs : s.Nonempty\nht : t.Nonempty\n⊢ ∀ x ∈ t, infEDist x s ≤ ENNReal.ofReal r",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\nr : ℝ\nhr : 0 ≤ r\nH1 : ∀ x ∈ s, infDist x t ≤ r\nH2 : ∀ x ∈ t, infDist x s ≤ r\nhs : s.Nonempty\nht : t.Nonempty\n⊢ ∀ x ∈ t, infEDist x s ≤ ENNReal.ofReal r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 599,
"column": 49
} | {
"line": 600,
"column": 60
} | {
"line": 602,
"column": 0
} | [
{
"pp": "δ : ℝ\nα : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : ProperSpace α\nE : Set α\nhE : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] E\nhδ : 0 ≤ δ\n⊢ cthickening δ E = ⋃ x ∈ E, closedBall x δ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
rw [cthickening_eq_biUnion_closedBall E hδ, hE.closure_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 43
} | {
"line": 113,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : MeasurableSpace α\nf : α →ₛ β\np : β → Prop\n⊢ (∃ y ∈ f.range, p y) ↔ ∃ x, p (f x)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"MeasureTheory.SimpleFunc",
"Membership.mem",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : MeasurableSpace α\nf : α →ₛ β\np : β → Prop\n⊢ (∃ y ∈ range ⇑f, p y) ↔ ∃ x, p (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 886,
"column": 2
} | {
"line": 886,
"column": 25
} | {
"line": 886,
"column": 26
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\ny : α\nr : ℝ\nh : y ∈ t\nH : hausdorffDist t s < r\nfin : hausdorffEDist t s ≠ ∞\n⊢ ∃ x ∈ s, dist x y < r",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instLT",
... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\ny : α\nr : ℝ\nh : y ∈ t\nH : hausdorffDist t s < r\nfin : hausdorffEDist t s ≠ ∞\n⊢ ∃ x ∈ s, dist y x < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 915,
"column": 2
} | {
"line": 915,
"column": 44
} | {
"line": 915,
"column": 45
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns t u : Set α\nfin : hausdorffEDist u t ≠ ∞\nI : hausdorffDist u s ≤ hausdorffDist u t + hausdorffDist t s\n⊢ hausdorffDist s u ≤ hausdorffDist s t + hausdorffDist t u",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"M... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns t u : Set α\nfin : hausdorffEDist u t ≠ ∞\nI : hausdorffDist u s ≤ hausdorffDist u t + hausdorffDist t s\n⊢ hausdorffDist s u ≤ hausdorffDist s t + hausdorffDist t u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 734,
"column": 2
} | {
"line": 737,
"column": 58
} | {
"line": 739,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : SemilatticeSup β\ninst✝ : OrderBot β\nf : γ → α →ₛ β\ns : Finset γ\na : α\n⊢ (s.sup f) a = s.sup fun c ↦ (f c) a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.sup_insert"... | [] | classical
refine Finset.induction_on s rfl ?_
intro a s _ ih
rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 734,
"column": 2
} | {
"line": 737,
"column": 58
} | {
"line": 739,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : SemilatticeSup β\ninst✝ : OrderBot β\nf : γ → α →ₛ β\ns : Finset γ\na : α\n⊢ (s.sup f) a = s.sup fun c ↦ (f c) a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.sup_insert"... | [] | classical
refine Finset.induction_on s rfl ?_
intro a s _ ih
rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 734,
"column": 2
} | {
"line": 737,
"column": 58
} | {
"line": 739,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : SemilatticeSup β\ninst✝ : OrderBot β\nf : γ → α →ₛ β\ns : Finset γ\na : α\n⊢ (s.sup f) a = s.sup fun c ↦ (f c) a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.sup_insert"... | [] | classical
refine Finset.induction_on s rfl ?_
intro a s _ ih
rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 798,
"column": 40
} | {
"line": 798,
"column": 91
} | {
"line": 798,
"column": 92
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ (f.restrict s).range\nh0 : r ≠ 0\nhs : MeasurableSet s\n⊢ r ∈ ⇑f '' s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.mem_image._simp_1",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ (f.restrict s).range\nh0 : r ≠ 0\nhs : MeasurableSet s\n⊢ ∃ x ∈ s, f x = r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 721,
"column": 42
} | {
"line": 721,
"column": 53
} | {
"line": 721,
"column": 54
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι), Measurabl... | [
"α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι), Measurable (f i)\ns :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 24,
"column": 34
} | {
"line": 24,
"column": 72
} | {
"line": 24,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : AddCommMonoid α\ninst✝³ : PartialOrder α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\nβ : Type u_2\nf : α → β\nk : α\nx : β\nx✝ : x ∈ f '' Ici k\ny : α\nhy : y ∈ Ici k\nhfy : f y = x\n⊢ (fun x ↦ f (x + k)) (y - k) = x",
"ppTerm": "?m.64",
"assi... | [
"α : Type u_1\ninst✝⁴ : AddCommMonoid α\ninst✝³ : PartialOrder α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\nβ : Type u_2\nf : α → β\nk : α\nx : β\nx✝ : x ∈ f '' Ici k\ny : α\nhy : y ∈ Ici k\nhfy : f y = x\n⊢ f y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 739,
"column": 6
} | {
"line": 739,
"column": 21
} | {
"line": 739,
"column": 22
} | [
{
"pp": "case pos\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι),... | [
"case pos\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι), Measurable ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 898,
"column": 2
} | {
"line": 907,
"column": 46
} | {
"line": 909,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ⨆ n, (eapprox f n) a = f a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.SimpleFunc.eapprox.eq_1",
"Rat.instOfNat",
"Eq.mpr",
"lt_of_le_of_lt",
"False"... | [] | rw [eapprox, iSup_approx_apply ennrealRatEmbed f a hf rfl]
refine le_antisymm (iSup_le fun i => iSup_le fun hi => hi) (le_of_not_gt ?_)
intro h
rcases ENNReal.lt_iff_exists_rat_btwn.1 h with ⟨q, _, lt_q, q_lt⟩
have :
(Real.toNNReal q : ℝ≥0∞) ≤ ⨆ (k : ℕ) (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k := by... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 898,
"column": 2
} | {
"line": 907,
"column": 46
} | {
"line": 909,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ⨆ n, (eapprox f n) a = f a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.SimpleFunc.eapprox.eq_1",
"Rat.instOfNat",
"Eq.mpr",
"lt_of_le_of_lt",
"False"... | [] | rw [eapprox, iSup_approx_apply ennrealRatEmbed f a hf rfl]
refine le_antisymm (iSup_le fun i => iSup_le fun hi => hi) (le_of_not_gt ?_)
intro h
rcases ENNReal.lt_iff_exists_rat_btwn.1 h with ⟨q, _, lt_q, q_lt⟩
have :
(Real.toNNReal q : ℝ≥0∞) ≤ ⨆ (k : ℕ) (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k := by... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 910,
"column": 2
} | {
"line": 910,
"column": 26
} | {
"line": 910,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ⨆ n, ⇑(eapprox f n) = f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"iSup",
"MeasureTheory.SimpleFunc",
"id",
"ConditionallyCompleteLinearOrder... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∀ (x : α), ⨆ i, (eapprox f i) x = f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 839,
"column": 79
} | {
"line": 839,
"column": 90
} | {
"line": 839,
"column": 91
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nh : ∀ x ∈ s, s ∈ 𝓝[>] x\nx₀ : α\nx₀s : x₀ ∈ s\nh₀ : IsTop x₀\nthis : s = {x₀} ∪ s \\ {x₀}\nx : α\nhx : x ∈ s \\ {x₀}... | [
"α : Type u_1\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nh : ∀ x ∈ s, s ∈ 𝓝[>] x\nx₀ : α\nx₀s : x₀ ∈ s\nh₀ : IsTop x₀\nthis : s = {x₀} ∪ s \\ {x₀}\nx : α\nhx : x ∈ s \\ {x₀}\n⊢ ¬x = x₀"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Real.Lemmas | {
"line": 105,
"column": 8
} | {
"line": 105,
"column": 81
} | {
"line": 105,
"column": 82
} | [
{
"pp": "case refine_1\ns : Set ℝ\nconn : s.OrdConnected\nnt : s.Nontrivial\nx : ℝ\nhx : x ∈ s\nε : ℝ\nε_pos : ε > 0\nz : ℝ\nhz : z ∈ s\nne : z ≠ x\nlt : z < x\nq : ℚ\nh₁ : z < ↑q ∧ x - ε < ↑q\nh₂ : ↑q < x\n⊢ |↑q - x| < ε",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IsRightCa... | [
"case refine_1\ns : Set ℝ\nconn : s.OrdConnected\nnt : s.Nontrivial\nx : ℝ\nhx : x ∈ s\nε : ℝ\nε_pos : ε > 0\nz : ℝ\nhz : z ∈ s\nne : z ≠ x\nlt : z < x\nq : ℚ\nh₁ : z < ↑q ∧ x - ε < ↑q\nh₂ : ↑q < x\n⊢ x - ε < ↑q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Real.Lemmas | {
"line": 109,
"column": 8
} | {
"line": 109,
"column": 72
} | {
"line": 109,
"column": 73
} | [
{
"pp": "case refine_2\ns : Set ℝ\nconn : s.OrdConnected\nnt : s.Nontrivial\nx : ℝ\nhx : x ∈ s\nε : ℝ\nε_pos : ε > 0\nz : ℝ\nhz : z ∈ s\nne : z ≠ x\nlt : x < z\nq : ℚ\nh₁ : x < ↑q\nh₂ : ↑q < z ∧ ↑q < x + ε\n⊢ |↑q - x| < ε",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"IsRightCa... | [
"case refine_2\ns : Set ℝ\nconn : s.OrdConnected\nnt : s.Nontrivial\nx : ℝ\nhx : x ∈ s\nε : ℝ\nε_pos : ε > 0\nz : ℝ\nhz : z ∈ s\nne : z ≠ x\nlt : x < z\nq : ℚ\nh₁ : x < ↑q\nh₂ : ↑q < z ∧ ↑q < x + ε\n⊢ ↑q < x + ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 864,
"column": 4
} | {
"line": 864,
"column": 25
} | {
"line": 865,
"column": 4
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhα : ... | [
"case h₁\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhα : Non... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1090,
"column": 2
} | {
"line": 1090,
"column": 76
} | {
"line": 1091,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α →ₛ ℝ≥0∞\nh : f ≤ g\n⊢ f.lintegral μ ≤ g.lintegral μ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.SimpleFunc.lintegral",
"PartialOrder.toPreorder",
"MeasureTheory.SimpleFunc",
"ENN... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf✝ g✝ : α →ₛ ℝ≥0∞\nh : f✝ ≤ g✝\nf g : α →ₛ ℝ≥0∞\n⊢ f.lintegral μ ≤ (f ⊔ g).lintegral μ"
] | refine Monotone.of_left_le_map_sup (f := (lintegral · μ)) (fun f g ↦ ?_) h | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 911,
"column": 4
} | {
"line": 911,
"column": 25
} | {
"line": 912,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (... | [] | simp [iSup_of_empty'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 911,
"column": 4
} | {
"line": 911,
"column": 25
} | {
"line": 912,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (... | [] | simp [iSup_of_empty'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 911,
"column": 4
} | {
"line": 911,
"column": 25
} | {
"line": 912,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (... | [] | simp [iSup_of_empty'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1210,
"column": 6
} | {
"line": 1210,
"column": 17
} | {
"line": 1210,
"column": 17
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_5\ninst✝ : AddZeroClass β\nf g : α →ₛ β\nhf : f.FinMeasSupp μ\nhg : g.FinMeasSupp μ\n⊢ (f + g).FinMeasSupp μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MeasureTheory.SimpleFunc.instAdd",
"Eq.mpr",
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_5\ninst✝ : AddZeroClass β\nf g : α →ₛ β\nhf : f.FinMeasSupp μ\nhg : g.FinMeasSupp μ\n⊢ (map (fun p ↦ p.1 + p.2) (f.pair g)).FinMeasSupp μ"
] | add_eq_map₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 1026,
"column": 2
} | {
"line": 1026,
"column": 79
} | {
"line": 1027,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_5\nι' : Type u_6\nf : ι → δ → α\nv : Filter ι\nh... | [
"case neg\nα : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_5\nι' : Type u_6\nf : ι → δ → α\nv : Filter ι\nhf : ∀ (i : ι... | let g : ℕ → Subtype p := Classical.choose (exists_surjective_nat (Subtype p)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 195,
"column": 15
} | {
"line": 195,
"column": 51
} | {
"line": 195,
"column": 52
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ≥0\nh : Measurable fun x ↦ ↑(f x)\n⊢ Measurable f",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ≥0\nh : Measurable fun x ↦ ↑(f x)\n⊢ Measurable f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 205,
"column": 14
} | {
"line": 205,
"column": 50
} | {
"line": 205,
"column": 51
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ≥0\nμ : Measure α\nh : AEMeasurable (fun x ↦ ↑(f x)) μ\n⊢ AEMeasurable f μ",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ≥0\nμ : Measure α\nh : AEMeasurable (fun x ↦ ↑(f x)) μ\n⊢ AEMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 27
} | {
"line": 119,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : PseudoMetrizableSpace β\ninst✝³ : MeasurableSpace β\ninst✝² : BorelSpace β\nι : Type u_3\ninst✝¹ : Nonempty ι\nμ : Measure α\nf : ι → α → β\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nhf : ∀ (n : ι), AEMeas... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : PseudoMetrizableSpace β\ninst✝³ : MeasurableSpace β\ninst✝² : BorelSpace β\nι : Type u_3\ninst✝¹ : Nonempty ι\nμ : Measure α\nf : ι → α → β\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nhf : ∀ (n : ι), AEMeasurable (f n)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 20
} | {
"line": 124,
"column": 21
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : PseudoMetrizableSpace β\ninst✝³ : MeasurableSpace β\ninst✝² : BorelSpace β\nι : Type u_3\ninst✝¹ : Nonempty ι\nμ : Measure α\nf : ι → α → β\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nhf : ∀ (n : ι), AEMeas... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : PseudoMetrizableSpace β\ninst✝³ : MeasurableSpace β\ninst✝² : BorelSpace β\nι : Type u_3\ninst✝¹ : Nonempty ι\nμ : Measure α\nf : ι → α → β\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nhf : ∀ (n : ι), AEMeasurable (f n)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 60
} | {
"line": 150,
"column": 61
} | [
{
"pp": "α : Type u_3\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_4\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nμ : Measure α\nh_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\nAs_mble : ∀ (i : ι), AEMeasurable ((As i).indicator fun x ↦ 1) μ\n⊢ ∀ᵐ (x : α) ∂μ, ... | [
"α : Type u_3\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_4\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nμ : Measure α\nh_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\nAs_mble : ∀ (i : ι), AEMeasurable ((As i).indicator fun x ↦ 1) μ\n⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 17
} | {
"line": 93,
"column": 4
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nN : ℕ\nihN : (nearestPtInd e N) x ≤ N\n⊢ (nearestPtInd e (N + 1)) x ≤ N + 1",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | | succ N ihN => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 17
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nN : ℕ\nihN : ∀ {k : ℕ}, k ≤ N → edist ((nearestPt e N) x) x ≤ edist (e k) x\nk : ℕ\nhk : k ≤ N + 1\n⊢ edist ((nearestPt e (N + 1)) x) x ≤ edist (e k) x",
"ppTerm": "... | [] | | succ N ihN => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 441,
"column": 42
} | {
"line": 441,
"column": 53
} | {
"line": 441,
"column": 54
} | [
{
"pp": "⊢ Measurable fun p ↦ (↑p).toENNReal",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"EReal.toENNReal_of_ne_top",
"ENNReal.ofReal",
"congrArg",
"EReal.coe_ne_top._simp_1",
"EReal.toENNReal",
"Measurable... | [
"⊢ Measurable fun p ↦ ENNReal.ofReal p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 517,
"column": 13
} | {
"line": 517,
"column": 25
} | {
"line": 517,
"column": 25
} | [
{
"pp": "case refine_4\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α✝\nmα✝ : MeasurableSpace α✝\nα : Type u_5\nβ : Type u_6\nγ : Type u_7\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\n⊢ Measurable fun r ↦ ↑r * ⊥",
"ppTerm": "?refine_4",
... | [
"case refine_4\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α✝\nmα✝ : MeasurableSpace α✝\nα : Type u_5\nβ : Type u_6\nγ : Type u_7\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\n⊢ Measurable fun r ↦ ⊥ * ↑r"
] | mul_comm _ ⊥ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 574,
"column": 2
} | {
"line": 574,
"column": 13
} | {
"line": 574,
"column": 14
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : IsFiniteMeasureOnCompacts μ\nb : ℝ\ns : Set ℝ≥0∞\nhs : s ∈ 𝓝 (μ {b})\n⊢ μ (Icc (b - 0) (b + 0)) ∈ s",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Icc_self",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",... | [
"μ : Measure ℝ\ninst✝ : IsFiniteMeasureOnCompacts μ\nb : ℝ\ns : Set ℝ≥0∞\nhs : s ∈ 𝓝 (μ {b})\n⊢ μ {b} ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 583,
"column": 4
} | {
"line": 583,
"column": 40
} | {
"line": 583,
"column": 41
} | [
{
"pp": "case right\nμ : Measure ℝ\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : NullSingletonClass μ\nb : ℝ\n⊢ Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[≥] 0) (𝓝 0)",
"ppTerm": "?right",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case right\nμ : Measure ℝ\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : NullSingletonClass μ\nb : ℝ\n⊢ Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[≥] 0) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 94
} | {
"line": 147,
"column": 6
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\nn : ℕ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ (a : α), g a ∂μ",
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nh₀ : ∫⁻ (a : α), f a ∂μ ≠ 0\nL : ℕ → ℝ≥0∞\nleft✝ : StrictMono L\nhLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)\nhL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))\nn : ℕ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ (a : α), g a ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 51
} | {
"line": 217,
"column": 52
} | [
{
"pp": "case pos\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Conti... | [
"case pos\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Continuous[instTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 220,
"column": 8
} | {
"line": 220,
"column": 23
} | {
"line": 220,
"column": 24
} | [
{
"pp": "case neg\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Conti... | [
"case neg\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Continuous[instTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 47
} | {
"line": 232,
"column": 48
} | [
{
"pp": "case pos\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Conti... | [
"case pos\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Continuous[instTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 41
} | {
"line": 438,
"column": 42
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl : List (α → M)\nhl : ∀ f ∈ l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x ↦ (List.map (fun f ↦ f x) l).prod) μ",
"ppTerm": "?m.23",
"assigned... | [
"α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl : List (α → M)\nhl : ∀ f ∈ l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x ↦ l.prod x) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 13
} | {
"line": 450,
"column": 14
} | [
{
"pp": "case mk\nα : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (Multiset.p... | [
"case mk\nα : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable l.prod μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 450,
"column": 46
} | {
"line": 450,
"column": 57
} | {
"line": 450,
"column": 58
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEStronglyMeasurable f μ\n⊢ ∀ f ∈ l, AEStronglyMeasurable f μ",
"... | [
"α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEStronglyMeasurable f μ\n⊢ ∀ f ∈ l, AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 45
} | {
"line": 456,
"column": 46
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\ns : Multiset (α → M)\nhs : ∀ f ∈ s, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x ↦ (Multiset.map (fun f ↦ f x) s).prod) μ",
"ppTerm": "?m.22",
... | [
"α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\ns : Multiset (α → M)\nhs : ∀ f ∈ s, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x ↦ s.prod x) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 469,
"column": 2
} | {
"line": 469,
"column": 40
} | {
"line": 469,
"column": 41
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nι : Type u_6\nf : ι → α → M\ns : Finset ι\nhf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ\n⊢ AEStronglyMeasurable (fun a ↦ ∏ i ∈ s, f i a) μ",
"ppTerm": "?m.24... | [
"α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nι : Type u_6\nf : ι → α → M\ns : Finset ι\nhf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ\n⊢ AEStronglyMeasurable (fun a ↦ (∏ c ∈ s, f c) a) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 74
} | {
"line": 638,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nmβ : MeasurableSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nf : α → β\nhf : AEStronglyMeasurable f μ\nt : Set β\nht1 : IsSeparable t\nht2 : ∀ᵐ (x : α) ∂μ, f x ∈ t\n⊢ (Measure.map f μ) (cl... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nmβ : MeasurableSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nf : α → β\nhf : AEStronglyMeasurable f μ\nt : Set β\nht1 : IsSeparable t\nht2 : ∀ᵐ (x : α) ∂μ, f x ∈ t\n⊢ ∀ᵐ (x : α) ∂μ, (⋂₀ {t_1 | IsClose... | refine ae_map_iff hf.aemeasurable isClosed_closure.measurableSet |>.2 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 224,
"column": 4
} | {
"line": 226,
"column": 74
} | {
"line": 227,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nh : ∀ᵐ (a : α) ∂μ, f a ≤ g a\nt : Set α\nhts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t\nht : MeasurableSet t\nht0 : μ t = 0\nthis : ∀ᵐ (x : α) ∂μ, x ∉ t\ns : α →ₛ ℝ≥0∞\nhfs : ⇑s ≤ fun a ↦ f a\na : α\n⊢ (s.restrict tᶜ) a ≤ (fun ... | [
"case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nh✝ : ∀ᵐ (a : α) ∂μ, f a ≤ g a\nt : Set α\nhts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t\nht : MeasurableSet t\nht0 : μ t = 0\nthis : ∀ᵐ (x : α) ∂μ, x ∉ t\ns : α →ₛ ℝ≥0∞\nhfs : ⇑s ≤ fun a ↦ f a\na : α\nh : a ∉ t\n⊢ s a ≤ g a"
] | by_cases h : a ∈ t <;>
simp only [restrict_apply s ht.compl, mem_compl_iff, h, not_true, not_false_eq_true,
indicator_of_notMem, zero_le, not_false_eq_true, indicator_of_mem] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 25
} | {
"line": 51,
"column": 4
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh_mono : Monotone f\nc : ℝ≥0 → ℝ≥0∞ := ofNNReal\nF : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a\ns : α →ₛ ℝ≥0\nhsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x\nr : ℝ≥0\nright✝ ha✝ : ↑r < 1\nha : r < 1\nrs : α →ₛ ℝ≥0 := Sim... | [
"case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh_mono : Monotone f\nc : ℝ≥0 → ℝ≥0∞ := ofNNReal\nF : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a\ns : α →ₛ ℝ≥0\nhsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x\nr : ℝ≥0\nright✝ ha✝ : ↑r < 1\nha : r < 1\nrs : α →ₛ ℝ≥0 := Simpl... | by_cases p_eq : p = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 31
} | {
"line": 197,
"column": 8
} | [
{
"pp": "case hbc\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nhf : StronglyMeasurable f\nc : ℝ\nx : α\nhfx : ‖f x‖ ≤ c\nh_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)\nhfx0 : f x = 0\nh_tendsto_norm : Tendsto (fun n ↦ ‖(hf.ap... | [
"case hbc\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nhf : StronglyMeasurable f\nc : ℝ\nx : α\nhfx : ‖f x‖ ≤ c\nh_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)\nhfx0 : f x = 0\nh_tendsto_norm : Tendsto (fun n ↦ ‖(hf.approx n) x‖) ... | · exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 42
} | {
"line": 224,
"column": 43
} | [
{
"pp": "case neg.inl\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nc : ℝ\nhf : StronglyMeasurable f\nhc : 0 ≤ c\nn : ℕ\nx : α\nh0 : ¬‖(hf.approx n) x‖ = 0\nh : ‖(hf.approx n) x‖ ≤ c\n⊢ ‖1‖ * ‖(hf.approx n) x‖ ≤ c",
"ppTerm": "?neg... | [
"case neg.inl\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nc : ℝ\nhf : StronglyMeasurable f\nhc : 0 ≤ c\nn : ℕ\nx : α\nh0 : ¬‖(hf.approx n) x‖ = 0\nh : ‖(hf.approx n) x‖ ≤ c\n⊢ ‖(hf.approx n) x‖ ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 840,
"column": 15
} | {
"line": 840,
"column": 47
} | {
"line": 840,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nf : α → β\nG : Type u_6\ninst✝² : Group G\ninst✝¹ : MulAction G β\ninst✝ : ContinuousConstSMul G β\nc : G\nh : AEStronglyMeasurable (fun x ↦ c • f x) μ\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.28",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nf : α → β\nG : Type u_6\ninst✝² : Group G\ninst✝¹ : MulAction G β\ninst✝ : ContinuousConstSMul G β\nc : G\nh : AEStronglyMeasurable (fun x ↦ c • f x) μ\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 45
} | {
"line": 53,
"column": 4
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\nε : ℝ≥0∞\n⊢ ε * μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\nε : ℝ≥0∞\n⊢ ε * μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 20
} | {
"line": 71,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), f a ≤ 1\nh'f : ∀ a ∈ sᶜ, f a = 0\nx : α\nhx : x ∈ s\n⊢ f x ≤ s.indicator 1 x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.indicat... | [
"case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), f a ≤ 1\nh'f : ∀ a ∈ sᶜ, f a = 0\nx : α\nhx : x ∈ s\n⊢ f x ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 20
} | {
"line": 72,
"column": 21
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), f a ≤ 1\nh'f : ∀ a ∈ sᶜ, f a = 0\nx : α\nhx : x ∉ s\n⊢ f x ≤ s.indicator 1 x",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq... | [
"case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), f a ≤ 1\nh'f : ∀ a ∈ sᶜ, f a = 0\nx : α\nhx : x ∉ s\n⊢ f x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 84
} | {
"line": 409,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nf : α → ℝ≥0∞\n⊢ ∫⁻ (a : α), f a ∂c • μ = c • ∫⁻ (a : α), f a ∂μ",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_def",
... | [] | simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.smul_iSup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 84
} | {
"line": 409,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nf : α → ℝ≥0∞\n⊢ ∫⁻ (a : α), f a ∂c • μ = c • ∫⁻ (a : α), f a ∂μ",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_def",
... | [] | simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.smul_iSup] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 84
} | {
"line": 409,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nf : α → ℝ≥0∞\n⊢ ∫⁻ (a : α), f a ∂c • μ = c • ∫⁻ (a : α), f a ∂μ",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_def",
... | [] | simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.smul_iSup] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 38
} | {
"line": 123,
"column": 4
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : ∫⁻ (x : α), f x ∂μ ≠ ∞\nhg : AEMeasurable g μ\nhgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢ Te... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : ∫⁻ (x : α), f x ∂μ ≠ ∞\nhg : AEMeasurable g μ\nhgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢ Tendsto (fun n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 13
} | {
"line": 141,
"column": 14
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhμ : ¬μ univ = 0\nhg : AEMeasurable g μ\nhfi : ∫⁻ (x : α), f x ∂μ ≠ ∞\nh : ∀ᵐ (x : α) ∂μ, f x < g x\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ univ → f x < g x",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhμ : ¬μ univ = 0\nhg : AEMeasurable g μ\nhfi : ∫⁻ (x : α), f x ∂μ ≠ ∞\nh : ∀ᵐ (x : α) ∂μ, f x < g x\n⊢ ∀ᵐ (x : α) ∂μ, f x < g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 17
} | {
"line": 493,
"column": 18
} | [
{
"pp": "case neg.e_a.e_6\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\ng : α →ₛ ℝ≥0∞\nhg : ⇑g ≤ fun a ↦ s.indicator f a\nthis : ⇑g ≤ f\nx : α\nH : x ∉ s\n⊢ g x = 0",
"ppTerm": "?neg.e_a.e_6✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"case neg.e_a.e_6\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\ng : α →ₛ ℝ≥0∞\nhg : ⇑g ≤ fun a ↦ s.indicator f a\nthis : ⇑g ≤ f\nx : α\nH : x ∉ s\n⊢ g x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 72,
"column": 14
} | {
"line": 74,
"column": 49
} | {
"line": 75,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nh_meas : ∀ (n : ℕ), Measurable (f n)\nh_mono✝ : ∀ (n : ℕ), f n.succ ≤ᵐ[μ] f n\nh_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞\nfn_le_f0 : ∫⁻ (a : α), ⨅ n, f n a ∂μ ≤ ∫⁻ (a : α), f 0 a ∂μ\nfn_le_f0' : ⨅ n, ∫⁻ (a : α), f n a ∂μ ≤ ∫⁻ (a : α), f 0... | [] | induction n with
| zero => rfl
| succ n ih => exact (h n).trans ih | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 72,
"column": 14
} | {
"line": 74,
"column": 49
} | {
"line": 75,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nh_meas : ∀ (n : ℕ), Measurable (f n)\nh_mono✝ : ∀ (n : ℕ), f n.succ ≤ᵐ[μ] f n\nh_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞\nfn_le_f0 : ∫⁻ (a : α), ⨅ n, f n a ∂μ ≤ ∫⁻ (a : α), f 0 a ∂μ\nfn_le_f0' : ⨅ n, ∫⁻ (a : α), f n a ∂μ ≤ ∫⁻ (a : α), f 0... | [] | induction n with
| zero => rfl
| succ n ih => exact (h n).trans ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 72,
"column": 14
} | {
"line": 74,
"column": 49
} | {
"line": 75,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nh_meas : ∀ (n : ℕ), Measurable (f n)\nh_mono✝ : ∀ (n : ℕ), f n.succ ≤ᵐ[μ] f n\nh_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞\nfn_le_f0 : ∫⁻ (a : α), ⨅ n, f n a ∂μ ≤ ∫⁻ (a : α), f 0 a ∂μ\nfn_le_f0' : ⨅ n, ∫⁻ (a : α), f n a ∂μ ≤ ∫⁻ (a : α), f 0... | [] | induction n with
| zero => rfl
| succ n ih => exact (h n).trans ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 55
} | {
"line": 166,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : ∫⁻ (a : α), f a ∂μ ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nhf₀ : ¬∫⁻ (a : α), f a ∂μ = 0\n⊢ ∃ g, ∃ (_ : ⇑g ≤ f), ∫⁻ (a : α), f a ∂μ - ε < g.lintegral μ",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"_private.Ma... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : ∫⁻ (a : α), f a ∂μ ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nhf₀ : ¬∫⁻ (a : α), f a ∂μ = 0\n⊢ ∫⁻ (a : α), f a ∂μ - ε < lintegral μ f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 550,
"column": 15
} | {
"line": 550,
"column": 47
} | {
"line": 550,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nG : Type u_6\ninst✝³ : TopologicalSpace β\ninst✝² : Group G\ninst✝¹ : MulAction G β\ninst✝ : ContinuousConstSMul G β\nm : MeasurableSpace α\nc : G\nh : StronglyMeasurable fun x ↦ c • f x\n⊢ StronglyMeasurable f",
"ppTerm": "?m.26",
"assigned": false,
"... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nG : Type u_6\ninst✝³ : TopologicalSpace β\ninst✝² : Group G\ninst✝¹ : MulAction G β\ninst✝ : ContinuousConstSMul G β\nm : MeasurableSpace α\nc : G\nh : StronglyMeasurable fun x ↦ c • f x\n⊢ StronglyMeasurable f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 41
} | {
"line": 619,
"column": 42
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl : ∀ f ∈ l, StronglyMeasurable f\n⊢ StronglyMeasurable fun x ↦ (List.map (fun f ↦ f x) l).prod",
"ppTerm": "?m.21",
"assigned": true,
"usedConstant... | [
"α : Type u_1\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl : ∀ f ∈ l, StronglyMeasurable f\n⊢ StronglyMeasurable fun x ↦ l.prod x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 632,
"column": 2
} | {
"line": 632,
"column": 13
} | {
"line": 632,
"column": 14
} | [
{
"pp": "case mk\nα : 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⊢ StronglyMeasurable (Multiset.prod (Quot.mk (⇑(List.i... | [
"case mk\nα : 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⊢ StronglyMeasurable l.prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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