module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 634,
"column": 15
} | {
"line": 634,
"column": 55
} | [
{
"pp": "β : Type u_5\nα : Type u_6\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace α\nf : α → β\nG : Type u_7\ninst✝² : Group G\ninst✝¹ : MulAction G β\ninst✝ : MeasurableConstSMul G β\nc : G\nh : Measurable fun x ↦ c • f x\n⊢ Measurable f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 639,
"column": 15
} | {
"line": 639,
"column": 55
} | [
{
"pp": "β : Type u_5\nα : Type u_6\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace α\nf : α → β\nμ : Measure α\nG : Type u_7\ninst✝² : Group G\ninst✝¹ : MulAction G β\ninst✝ : MeasurableConstSMul G β\nc : G\nh : AEMeasurable (fun x ↦ c • f x) μ\n⊢ AEMeasurable f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 735,
"column": 6
} | {
"line": 735,
"column": 40
} | [
{
"pp": "α✝ : Type u_1\nM : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace M\ninst✝⁴ : MeasurableSpace α\ninst✝³ : SMul M α\ninst✝² : SMul Mᵐᵒᵖ α\ninst✝¹ : IsCentralScalar M α\ninst✝ : MeasurableSMul M α\nx : α\n⊢ Measurable fun c ↦ MulOpposite.op (unop c) • x",
"usedConstants": [
"Eq.mpr",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 790,
"column": 2
} | {
"line": 790,
"column": 41
} | [
{
"pp": "M : Type u_2\nα : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nl : List (α → M)\nhl : ∀ f ∈ l, Measurable f\n⊢ Measurable fun x ↦ (map (fun f ↦ f x) l).prod",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 795,
"column": 2
} | {
"line": 795,
"column": 41
} | [
{
"pp": "M : Type u_2\nα : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\nl : List (α → M)\nhl : ∀ f ∈ l, AEMeasurable f μ\n⊢ AEMeasurable (fun x ↦ (map (fun f ↦ f x) l).prod) μ",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 808,
"column": 2
} | {
"line": 808,
"column": 13
} | [
{
"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\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, Measurable f\n⊢ Measurable (prod (Quot.mk (⇑(List.isSetoid (α → M))) l))",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 808,
"column": 36
} | {
"line": 808,
"column": 47
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 814,
"column": 2
} | {
"line": 814,
"column": 13
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 814,
"column": 38
} | {
"line": 814,
"column": 49
} | [
{
"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",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 45
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Multiset.map",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 824,
"column": 2
} | {
"line": 824,
"column": 45
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Meas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 856,
"column": 2
} | {
"line": 856,
"column": 40
} | [
{
"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",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 74,
"column": 23
} | {
"line": 74,
"column": 65
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nh : ∀ (x y : α), dist (f x) (f y) ≤ dist x y\n⊢ ∀ (x y : α), dist (f x) (f y) ≤ ↑1 * dist x y",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 86,
"column": 28
} | {
"line": 86,
"column": 64
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nf : α → ℝ\nK : ℝ≥0\nh : ∀ (x y : α), f x ≤ f y + ↑K * dist x y\n⊢ LipschitzWith K f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 35
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : LipschitzWith K f\nx : α\nn : ℕ\n⊢ dist (f^[n] x) ((f^[n] ∘ f) x) ≤ ↑K ^ n * dist x (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 80
} | [
{
"pp": "x y : ℝ\n⊢ dist x.toNNReal y.toNNReal ≤ ↑1 * dist x y",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.lattice",
"abs",
"congrArg",
"Real.instSub",
"HSub.hSub",
"Real.semiring",
"id",
"NNReal",
"Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 37
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf g : α → ℝ\nKf Kg : ℝ≥0\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n⊢ LipschitzWith (max Kf Kg) fun x ↦ max (f x) (g x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 37
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf g : α → ℝ\nKf Kg : ℝ≥0\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n⊢ LipschitzWith (max Kf Kg) fun x ↦ min (f x) (g x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 13
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ max (f x) a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 29
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ max a (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 13
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ min (f x) a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 29
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ min a (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 225,
"column": 23
} | {
"line": 225,
"column": 65
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns : Set α\nf : α → β\nh : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y\n⊢ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ ↑1 * dist x y",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 38
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nf : α → ℝ\nK : ℝ≥0\nh : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + ↑K * dist x y\n⊢ LipschitzOnWith K f s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 24
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nhf : LocallyLipschitz f\na : ℝ\n⊢ LocallyLipschitz fun x ↦ max a (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 24
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nhf : LocallyLipschitz f\na : ℝ\n⊢ LocallyLipschitz fun x ↦ min a (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 191,
"column": 3
} | {
"line": 191,
"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)",
"usedConstants": [
"BorelSpace.measurable_eq",... | by borelize α; symm; apply borel_comap | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 68
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nf : α → ℝ\ns : Set α\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nhs : s.Nonempty\nthis : Nonempty ↑s\ng : α → ℝ := fun y ↦ ⨅ x, f ↑x + ↑K * dist y ↑x\nB : ∀ (y : α), BddBelow (range fun x ↦ f ↑x + ↑K * dist y ↑x)\nx : α\nhx : x ∈ s\n⊢ g x ≤ f x",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 343,
"column": 6
} | {
"line": 343,
"column": 35
} | [
{
"pp": "case h\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✝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 : Topo... | inseparable_iff_forall_isOpen | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 418,
"column": 6
} | {
"line": 418,
"column": 27
} | [
{
"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... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 147,
"column": 37
} | {
"line": 147,
"column": 66
} | [
{
"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",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"False",
"ENNReal.ofNNRea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 157,
"column": 57
} | {
"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... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 439,
"column": 6
} | {
"line": 439,
"column": 27
} | [
{
"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... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 217,
"column": 24
} | {
"line": 217,
"column": 35
} | [
{
"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",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 219,
"column": 37
} | {
"line": 219,
"column": 66
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 312,
"column": 39
} | {
"line": 312,
"column": 68
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"False",
"ENNReal.ofNNReal",
"instHDiv",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 325,
"column": 30
} | {
"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 ... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 349,
"column": 4
} | {
"line": 353,
"column": 100
} | [
{
"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",
"usedConstants": [
"ENNReal.instAdd",
"le_refl",
"Trans.trans",
"ENNReal.instAddCommMonoid",
"covariant_swap_add_of_covariant_add",
... | 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.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 653,
"column": 55
} | {
"line": 653,
"column": 85
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 13
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nx : α\nxs : x ∈ s\nthis : infEDist x ∅ ≤ hausdorffEDist s ∅\n⊢ hausdorffEDist s ∅ = ∞",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 590,
"column": 2
} | {
"line": 591,
"column": 9
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"iInf",
"congrArg",
"Metric.infDist",
"Membership.mem",
"Set.Elem",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 744,
"column": 2
} | {
"line": 744,
"column": 36
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 64,
"column": 19
} | {
"line": 64,
"column": 55
} | [
{
"pp": "M : Type u\nG : Type v\nX : Type w\ninst✝⁴ : PseudoEMetricSpace X\ninst✝³ : SMul M X\ninst✝² : SMul Mᵐᵒᵖ X\ninst✝¹ : IsCentralScalar M X\ninst✝ : IsIsometricSMul M X\nc : Mᵐᵒᵖ\nx y : X\n⊢ edist ((fun x ↦ c • x) x) ((fun x ↦ c • x) y) = edist x y",
"usedConstants": [
"PseudoEMetricSpace.toWeak... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 836,
"column": 6
} | {
"line": 836,
"column": 91
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 837,
"column": 6
} | {
"line": 837,
"column": 91
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 451,
"column": 19
} | {
"line": 451,
"column": 35
} | [
{
"pp": "M : Type u\nG : Type v\nX : Type w\nY : Type u_1\ninst✝³ : PseudoEMetricSpace X\ninst✝² : PseudoEMetricSpace Y\ninst✝¹ : SMul M X\ninst✝ : IsIsometricSMul M X\nc : M\nx y : Xᵐᵒᵖ\n⊢ edist ((fun x ↦ c • x) x) ((fun x ↦ c • x) y) = edist x y",
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoE... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 455,
"column": 15
} | {
"line": 455,
"column": 31
} | [
{
"pp": "M : Type u\nG : Type v\nX : Type w\nY : Type u_1\ninst✝³ : PseudoEMetricSpace X\ninst✝² : PseudoEMetricSpace Y\ninst✝¹ : SMul M X\ninst✝ : IsIsometricSMul M X\nc : ULift.{u_2, u} M\n⊢ Isometry fun x ↦ c • x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 459,
"column": 19
} | {
"line": 459,
"column": 35
} | [
{
"pp": "M : Type u\nG : Type v\nX : Type w\nY : Type u_1\ninst✝³ : PseudoEMetricSpace X\ninst✝² : PseudoEMetricSpace Y\ninst✝¹ : SMul M X\ninst✝ : IsIsometricSMul M X\nc : M\nx y : ULift.{u_2, w} X\n⊢ edist ((fun x ↦ c • x) x) ((fun x ↦ c • x) y) = edist x y",
"usedConstants": [
"PseudoEMetricSpace.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 45
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.toLT",
"Iff.of_eq",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 886,
"column": 2
} | {
"line": 886,
"column": 25
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instLT",
"Membership.mem",
"Exists",
"id... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 915,
"column": 2
} | {
"line": 915,
"column": 44
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Metric.hausdorffDist",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 35
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns : Set α\n⊢ μ s ≤ μ ((fun x ↦ c • x) ⁻¹' s)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 38
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns : Set α\n⊢ μ (c • s) = μ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 23
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s ∩ t) = μ (s ∩ c • t)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 23
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s ∪ t) = μ (s ∪ c • t)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 23
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s \\ t) = μ (s \\ c • t)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 23
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ ((c⁻¹ • s) ∆ t) = μ (s ∆ (c • t))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 63
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nH : μ.InnerRegularWRT p q\nc : ℝ≥0∞\nU : Set α\nhU : q U\nr : ℝ≥0∞\nhr : r < c * ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K\n⊢ ∃ K ⊆ U, p K ∧ r < (c • μ) K",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 45
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.toLT",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 423,
"column": 4
} | {
"line": 423,
"column": 88
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\ninst✝ : μ.OuterRegular\nx : ℝ≥0∞\nhx : x ≠ ∞\nh0 : x ≠ 0\nA : Set α\nx✝ : MeasurableSet A\nr : ℝ≥0∞\nhr : r > x * ⨅ U, ⨅ (_ : A ⊆ U), ⨅ (_ : IsOpen[inst✝¹] U), μ U\n⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ (x • μ) U < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 307,
"column": 2
} | {
"line": 307,
"column": 40
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\nμ : Measure α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : MeasurableConstSMul G α\ns : Set α\nhs : NullMeasurableSet s μ\nc : G\n⊢ NullMeasurableSet (c • s) μ",
"usedConstants": [
"Eq.mpr",
"instH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 124,
"column": 17
} | {
"line": 124,
"column": 28
} | [
{
"pp": "case singleton\nL : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\na✝ : ι\nhs : ∀ i ∈ {a✝}, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto ({a✝}.sup' ⋯ f) l (𝓝 ({a✝}.sup' ⋯ g))",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 126,
"column": 8
} | {
"line": 126,
"column": 23
} | [
{
"pp": "case cons\nL : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns✝ : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\na : ι\ns : Finset ι\nha : a ∉ s\nhne : s.Nonempty\nihs : (∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) → Tendsto (s.sup' h... | forall_mem_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Lattice | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 40
} | [
{
"pp": "L : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\nhne : s.Nonempty\nhs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto (fun a ↦ s.sup' hne fun x ↦ f x a) l (𝓝 (s.sup' hne g))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 15
} | [
{
"pp": "case inl\nL : Type u_1\ninst✝³ : TopologicalSpace L\nι : Type u_3\nα : Type u_4\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\nhs : ∀ i ∈ ∅, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto (∅.sup f) l (𝓝 (∅.sup g))",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 39
} | [
{
"pp": "L : Type u_1\ninst✝³ : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\nhs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto (fun a ↦ s.sup fun x ↦ f x a) l (𝓝 (s.sup g))",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 40
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.sup' hne f) x",
"usedConstants": []
}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 40
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.sup' hne f) t x",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 39
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nx : X\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.sup f) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 39
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.sup f) t x",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 40
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeInf L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.inf' hne f) x",
"usedConstants": []
}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 40
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeInf L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.inf' hne f) t x",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 39
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeInf L\ninst✝¹ : OrderTop L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nx : X\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.inf f) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 39
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeInf L\ninst✝¹ : OrderTop L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.inf f) t x",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 586,
"column": 4
} | {
"line": 586,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ : Measure α\np : Set α → Prop\ninst✝¹ : TopologicalSpace α\ninst✝ : μ.OuterRegular\nH : μ.InnerRegularWRT p IsOpen[inst✝¹]\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen[inst✝¹] U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nr : ℝ≥0∞\nhr : r < μ s\nthis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 46
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ns : Set α\nh : s.Countable\nμ : Measure α\ninst✝ : NoAtoms μ\n⊢ ∀ᵐ (x : α) ∂μ, x ∉ s",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Filter.Eventually",
"setOf",
"Membership.mem"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 63
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ns : Set α\nh : s.Countable\nμ : Measure α\ninst✝ : NoAtoms μ\n⊢ ∀ᵐ (x : α) ∂μ, x ∉ s",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Filter.Eventually",
"setOf",
"Membership.mem"... | simpa only [ae_iff, Classical.not_not] using h.measure_zero μ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 63
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ns : Set α\nh : s.Countable\nμ : Measure α\ninst✝ : NoAtoms μ\n⊢ ∀ᵐ (x : α) ∂μ, x ∉ s",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Filter.Eventually",
"setOf",
"Membership.mem"... | simpa only [ae_iff, Classical.not_not] using h.measure_zero μ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 63
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ns : Set α\nh : s.Countable\nμ : Measure α\ninst✝ : NoAtoms μ\n⊢ ∀ᵐ (x : α) ∂μ, x ∉ s",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Filter.Eventually",
"setOf",
"Membership.mem"... | simpa only [ae_iff, Classical.not_not] using h.measure_zero μ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.Real.Lemmas | {
"line": 105,
"column": 8
} | {
"line": 105,
"column": 81
} | [
{
"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",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Real.Lemmas | {
"line": 109,
"column": 8
} | {
"line": 109,
"column": 72
} | [
{
"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",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 660,
"column": 49
} | {
"line": 660,
"column": 67
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nH✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]\nhfin : ∀ {s : Set α}, μ s ≠ ∞\ns : ℕ → Set α\nhsd : Pairwise (Function.onFun Disjoint s)\nhsm : ∀ (i : ℕ), Meas... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 33,
"column": 34
} | {
"line": 33,
"column": 72
} | [
{
"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",
"usedConstants": [
"Eq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 195,
"column": 15
} | {
"line": 195,
"column": 51
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ≥0\nh : Measurable fun x ↦ ↑(f x)\n⊢ Measurable f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 205,
"column": 14
} | {
"line": 205,
"column": 50
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 70,
"column": 10
} | {
"line": 70,
"column": 46
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 917,
"column": 4
} | {
"line": 917,
"column": 64
} | [
{
"pp": "case a.H\nα : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : μ.InnerRegularCompactLTTop\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : R1Space α\ninst✝ : BorelSpace α\nK : Set α\nhK : IsCompact K\n⊢ ∀ (c : ℝ≥0∞), μ K < c → ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen[inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 418,
"column": 42
} | {
"line": 418,
"column": 53
} | [
{
"pp": "⊢ Measurable fun p ↦ (↑p).toReal",
"usedConstants": [
"Real",
"Measurable",
"id",
"Real.measurableSpace",
"EReal.toReal",
"Real.toEReal"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 444,
"column": 42
} | {
"line": 444,
"column": 53
} | [
{
"pp": "⊢ Measurable fun p ↦ (↑p).toENNReal",
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"EReal.toENNReal_of_ne_top",
"ENNReal.ofReal",
"congrArg",
"EReal.coe_ne_top._simp_1",
"EReal.toENNReal",
"Measurable",
"EReal",
"ENNReal.measurableS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 534,
"column": 13
} | {
"line": 534,
"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 * ⊥",
"usedConstants": [
"C... | mul_comm _ ⊥ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 52
} | [
{
"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}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 591,
"column": 2
} | {
"line": 591,
"column": 13
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Set.Icc_self",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"Real.instZero",
"Real.instAddMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 600,
"column": 4
} | {
"line": 600,
"column": 40
} | [
{
"pp": "case right\nμ : Measure ℝ\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : NoAtoms μ\nb : ℝ\n⊢ Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[≥] 0) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 358,
"column": 85
} | {
"line": 388,
"column": 61
} | [
{
"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": 400,
"column": 2
} | {
"line": 400,
"column": 52
} | [
{
"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}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 52
} | [
{
"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}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 25
} | [
{
"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... | by_cases p_eq : p = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 94
} | [
{
"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",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 591,
"column": 2
} | {
"line": 591,
"column": 29
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : OpensMeasurableSpace α\nmδ : MeasurableSpace δ\ninst✝² : LinearOrder α\ninst✝¹ : SecondCountableTopology α\ninst✝ : OrderClosedTopology α\nf g : δ → α\nhf : Measurable f\nhg : Measurable g\n⊢ Measurable fun a ↦ Ma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 602,
"column": 2
} | {
"line": 602,
"column": 28
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : OpensMeasurableSpace α\nmδ : MeasurableSpace δ\ninst✝² : LinearOrder α\ninst✝¹ : SecondCountableTopology α\ninst✝ : OrderClosedTopology α\nf g : δ → α\nhf : Measurable f\nhg : Measurable g\n⊢ Measurable fun a ↦ Mi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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