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.OuterMeasure.Caratheodory | {
"line": 115,
"column": 2
} | {
"line": 116,
"column": 25
} | [
{
"pp": "α : Type u\nm : OuterMeasure α\nι : Type u_1\ninst✝¹ : Preorder ι\ninst✝ : LocallyFiniteOrderBot ι\ns : ι → Set α\nh : ∀ (i : ι), m.IsCaratheodory (s i)\ni : ι\n⊢ m.IsCaratheodory ((partialSups s) i)",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"_private.Mathlib.Meas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 15
} | [
{
"pp": "α : Type u\nm : OuterMeasure α\ns : ℕ → Set α\nh : ∀ (i : ℕ), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nn : ℕ\na : α\n⊢ a ∈ s n ∩ ⋃ k, ⋃ (_ : k < n), s k → a ∈ ∅",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"False",
"Set.mem_empty_iff_false._s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 45
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nr C : ℝ≥0∞\nf : ℕ → α\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\n⊢ edist (f 0) a ≤ C / (1 - r)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nf : ℕ → α\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\n⊢ edist (f 0) a ≤ 2 * C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 15
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nf : ℕ → α\nhr : r < 1\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\nleft✝ : 0 < C\nr₀ : 0 ≤ r\n⊢ HasSum (HPow.hPow r) (1 - r)⁻¹",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 387,
"column": 6
} | {
"line": 389,
"column": 27
} | [
{
"pp": "case h.refine_2.inr\nα : Type u_3\nι : Type u_4\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ {univ}\n⊢ ↑∅ ⊆ {i} ∧ ∃ f, (∀ x ∈ ∅, f x ∈ π x) ∧ s = ⋂ x ∈ ∅, f x",
"usedConstants": [
"Eq.mpr",
"False",
"Finset.coe_empty",
"Iff.of_eq",
"congrArg",
"Set.iInter",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 684,
"column": 78
} | {
"line": 684,
"column": 95
} | [
{
"pp": "k : ℕ\nhn : 0 < k.succ\n⊢ ↑(∏ i ∈ Finset.range k.succ, (i + 1)) * (∏ _k ∈ Finset.range k.succ, ↑k.succ)⁻¹ ≤ 1 / ↑k.succ",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul... | ← inv_eq_one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 740,
"column": 2
} | {
"line": 740,
"column": 13
} | [
{
"pp": "R : Type u_4\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\n⊢ Tendsto (fun x ↦ ↑⌊x⌋₊ / x) atTop (𝓝 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 756,
"column": 2
} | {
"line": 756,
"column": 13
} | [
{
"pp": "R : Type u_4\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\n⊢ Tendsto (fun x ↦ ↑⌈x⌉₊ / x) atTop (𝓝 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Induced | {
"line": 387,
"column": 58
} | {
"line": 387,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t\nhs✝ : ¬ms = ∞\nr : ℝ≥0∞\nhs : r > ms\n⊢ ∃ t, MeasurableSet t ∧ s ⊆ t ∧ m t < r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 537,
"column": 15
} | {
"line": 537,
"column": 26
} | [
{
"pp": "α : Type u_3\nd : DynkinSystem α\na : Set α\nh : d.Has aᶜ\n⊢ d.Has a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 539,
"column": 36
} | {
"line": 539,
"column": 47
} | [
{
"pp": "α : Type u_3\nd : DynkinSystem α\n⊢ d.Has univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.AE | {
"line": 128,
"column": 26
} | {
"line": 128,
"column": 42
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\nh : ∀ (s : Set α), μ sᶜ = 0 ↔ s = univ\na : α\nha : μ {a} = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nm : OuterMeasure α\n⊢ m ≤ μ.toOuterMeasure ↔ ∀ (s : Set α), MeasurableSet s → m s ≤ μ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 38
} | [
{
"pp": "α : Type u_1\nι : Sort u_5\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable ι\nμ : ι → Measure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), (μ i) t = (μ i) s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 52
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ ν : Measure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s ∧ ν t = ν s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 33
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\nh's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)\n⊢ μ h's.choose = μ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 34
} | [
{
"pp": "case refine_2\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhtm : MeasurableSet t\nhst : s =ᶠ[ae μ] t\nthis : toMeasurable μ (s \\ t) =ᶠ[ae μ] ∅\n⊢ t ∪ toMeasurable μ (s \\ t) =ᶠ[ae μ] s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Monotone | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 51
} | [
{
"pp": "case neg.inl\nα : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : OrderBot α\nn : α\nφ : α → α\nk : α\nhφ : StrictMonoOn φ (Iic (succ k))\nhk : ¬IsMax k\nhm : succ k ≤ succ k\nih : k ≤ φ k\n⊢ succ (φ k) ≤ φ (succ k)",
"usedConstants": [
"le_rfl",
... | refine succ_le_of_lt (hφ (le_succ _) le_rfl ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.AEDisjoint | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 17
} | [
{
"pp": "α : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh : AEDisjoint μ s t\nx : α\nhx : x ∈ s \\ toMeasurable μ (s ∩ t)\n⊢ x ∈ s \\ t",
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"id",
"SDiff.sdiff",
"And",
"Set.mem_diff._simp_1",
"Eq",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 18
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : μ sᶜ = 0\n⊢ μ s = μ univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | {
"line": 89,
"column": 41
} | {
"line": 89,
"column": 61
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nh : μ ≪ ν\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ (map f ν) s = 0 → (map f μ) s = 0",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Mea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | {
"line": 112,
"column": 65
} | {
"line": 112,
"column": 76
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ₁ μ₂ ν : Measure α\nh : μ₁ + μ₂ ≪ ν\ns : Set α\nhs0 : ν s = 0\n⊢ μ₁ s = 0 ∧ μ₂ s = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.BorelCantelli | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 57
} | [
{
"pp": "case h.e_6.h.h\nα : Type u_1\nι : Type u_2\nF : Type u_3\ninst✝² : FunLike F (Set α) ℝ≥0∞\ninst✝¹ : OuterMeasureClass F α\ninst✝ : Countable ι\nμ : F\ns : ι → Set α\nh : ∑' (i : ι), μ (s i) ≠ ∞\nx : α\n⊢ x ∈ {a | ¬{i | a ∈ s i}.Finite} ↔ x ∈ limsup s cofinite",
"usedConstants": [
"Classical.n... | simp [mem_limsup_iff_frequently_mem, Filter.Frequently] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.OuterMeasure.BorelCantelli | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 23
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\np : ℕ → α → Prop\nhp : ∑' (i : ℕ), μ {x | p i x} ≠ ∞\n⊢ μ {x | ∃ᶠ (n : ℕ) in atTop, p n x} = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
"setOf",
"instArch... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Map | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 9
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf g : α → β\nhf : Measurable f\nhg : Measurable g\nh : f =ᶠ[ae μ] g\ns : Set β\nhs : MeasurableSet s\n⊢ ((mapₗ f) μ) s = ((mapₗ g) μ) s",
"usedConstants": [
"MeasureTheory.Measure.mapₗ",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Map | {
"line": 123,
"column": 38
} | {
"line": 123,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf g : α → β\nh : f =ᶠ[ae μ] g\nhf : ¬AEMeasurable f μ\n⊢ ¬AEMeasurable g μ",
"usedConstants": [
"Eq.mpr",
"AEMeasurable",
"congrArg",
"id",
"aemeasurable_congr",
"propext",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Map | {
"line": 129,
"column": 54
} | {
"line": 129,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nμ : Measure α\nf : α → β\nthis : ∀ (c : ℝ≥0∞), map f (c • μ) = c • map f μ\n⊢ map f (c • μ) = c • map f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 46
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f μ μ\nhs : f ⁻¹' s =ᵐ[μ] s\nn : ℕ\n⊢ (fun n ↦ (preimage f)^[n] s) n =ᵐ[μ] s",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"Filter.EventuallyEq",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 46
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f μ μ\nhs : f ⁻¹' s =ᵐ[μ] s\nn : ℕ\n⊢ (fun n ↦ (preimage f)^[n] s) n =ᵐ[μ] s",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"Filter.EventuallyEq",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 40
} | [
{
"pp": "G : Type u_5\nα : Type u_6\ninst✝¹ : Group G\ninst✝ : MulAction G α\nx✝ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ng : G\nh_qmp : QuasiMeasurePreserving (fun x ↦ g⁻¹ • x) μ μ\nh_ae_eq : s =ᵐ[μ] t\n⊢ g • s =ᵐ[μ] g • t",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"instHS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Map | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx✝ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ne : α ≃ᵐ β\nμ₁ μ₂ : Measure α\nhμ : map (⇑e.symm) (map (⇑e) μ₁) = map (⇑e.symm) (map (⇑e) μ₂)\n⊢ μ₁ = μ₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nμ : Measure β\nc : ℝ≥0∞\nhc : c ≠ 0\nh : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ\ns : Set α\nhs : MeasurableSet s\n⊢ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) (c ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nμ : Measure β\nc : ℝ≥0∞\nhc : c ≠ 0\nh : ¬(Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ)\n⊢ ¬(Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) (c • μ))",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 45,
"column": 6
} | {
"line": 45,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\nmβ : MeasurableSpace β\nμ✝ ν ν₁ ν₂ : Measure α\ns t : Set α\nμ : Measure α\nhs : Fact (μ s < ∞)\n⊢ (μ.restrict s) univ < ∞",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.toLT",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 141,
"column": 2
} | {
"line": 142,
"column": 9
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : NullMeasurableSet s μ\nht : NullMeasurableSet t μ\n⊢ μ (s ∆ t) = μ (s \\ t) + μ (t \\ s)",
"usedConstants": [
"ENNReal.instAdd",
"MeasureTheory.Measure",
"Lattice.toSemilatticeSup",
"CompleteLattice.toCond... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 54
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nhμ : measureUnivNNReal μ ≤ 0\n⊢ μ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 69,
"column": 18
} | {
"line": 69,
"column": 29
} | [
{
"pp": "X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\nb : X\nhsne : s.Nonempty\nthis : Nonempty ↑s\nhsub : ¬s ⊆ Iio b\nh : Tendsto Subtype.val atTop (𝓝[<] b)\na : X\nhas : a ∈ s\nha : a ∉ Iio b\nc : X\nhcs : c ∈ s\nhcb : ↑⟨c, hcs⟩ ∈ Iio b\nhac : ⟨a, has... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 70,
"column": 18
} | {
"line": 70,
"column": 29
} | [
{
"pp": "X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\nb : X\nhsne : s.Nonempty\nthis : Nonempty ↑s\nhsub : ¬s ⊆ Iio b\nh : Tendsto Subtype.val atTop (𝓝[<] b)\na : X\nhas : a ∈ s\nha : a ∉ Iio b\nc : X\nhcs : c ∈ s\nhcb : ↑⟨c, hcs⟩ ∈ Iio b\nhac : ⟨a, has... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 71,
"column": 18
} | {
"line": 71,
"column": 29
} | [
{
"pp": "X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\nb : X\nhsne : s.Nonempty\nthis : Nonempty ↑s\nhsub : ¬s ⊆ Iio b\nh : Tendsto Subtype.val atTop (𝓝[<] b)\na : X\nhas : a ∈ s\nha : a ∉ Iio b\nc : X\nhcs : c ∈ s\nhcb : ↑⟨c, hcs⟩ ∈ Iio b\nhac : ⟨a, has... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 32
} | [
{
"pp": "X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\na : X\nhsa : s ⊆ Ioi a\nhs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s\nha : IsPredPrelimit a\nb' : Xᵒᵈ\nhb' : b' < toDual a\n⊢ ∃ a_1 < toDual a, Ioo a_1 (toDual a) ⊆ ⇑ofDual ⁻¹' s",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 30
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nx : α\n⊢ ∃ s, x ∈ s ∧ IsOpen[inst✝¹] s ∧ μ s < ∞",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 316,
"column": 17
} | {
"line": 316,
"column": 51
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh : s ∪ t =ᶠ[ae μ] s\n⊢ μ (t \\ s) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 318,
"column": 28
} | {
"line": 318,
"column": 43
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh : μ (t \\ s) = 0\n⊢ μ ((s ∪ t) \\ s) = 0",
"usedConstants": [
"Eq.mpr",
"Set.union_diff_left",
"MeasureTheory.Measure",
"congrArg",
"Set.instUnion",
"id",
"SDiff.sdiff",
"ENNReal",
... | union_diff_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 60
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh₁ : s ≤ᶠ[ae μ] t\nh₂ : μ t ≤ μ s\nhsm : NullMeasurableSet s μ\nht : μ t ≠ ∞\n⊢ μ (t \\ s) = 0",
"usedConstants": [
"MeasureTheory.Measure",
"LE.le.antisymm",
"MeasureTheory.measure_mono_ae",
"ENNReal",
"... | replace h₂ : μ t = μ s := h₂.antisymm (measure_mono_ae h₁) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ ν : Measure α\ns t : Set α\na : α\ninst✝¹ : SFinite μ\ninst✝ : SFinite ν\nthis : ∀ (b : Bool), SFinite (bif b then μ else ν)\n⊢ SFinite (μ + ν)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 24
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nc : ℕ → ℝ≥0\nhc₀ : ∀ (i : ℕ), 0 < c i\nhc : ∑' (i : ℕ), (sfiniteSeq μ i) univ * ↑(c i) < ∞\nthis : ∀ {s : Set α}, (sum fun n ↦ c n • sfiniteSeq μ n) s = 0 ↔ μ s = 0\n⊢ (sum fun n ↦ c n • sfiniteSeq μ n) univ < ∞",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 44
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nC : Set (Set α)\nhC : IsCountablySpanning C\nhm : C ⊆ MeasurableSet\nht : ∀ t ∈ C, (μ.restrict t) s = 0\nt : Set α\nhtc : t ∈ C\n⊢ μ (s ∩ t) = 0",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 34
} | [
{
"pp": "α : Type u_2\n_m0 : MeasurableSpace α\nR : Type u_7\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nμ : Measure α\ns : Set α\n⊢ (c • μ).restrict s = c • μ.restrict s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsCompact s\n⊢ (∀ a ∈ s, ∃ t ∈ 𝓝[s] a, μ t = 0) → μ s = 0",
"usedConstants": [
"Filter.instMembership",
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 53
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nt : Set α\n⊢ μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Set.instUnion",
"MeasureTheory.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 46
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns s' t : Set α\nht : MeasurableSet t\n⊢ (μ.restrict (s ∪ s')) t ≤ (μ.restrict s + μ.restrict s') t",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.restric... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 473,
"column": 6
} | {
"line": 479,
"column": 45
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nt : ℕ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊆ x2) t\nT : ℕ → Set α := fun n ↦ toMeasurable μ (t n)\nTd : ℕ → Set α := disjointed T\nhm : ∀ (n : ℕ), MeasurableSet (Td n)\nI : Finset ℕ\nN : ℕ\nhN : ∀ i ∈ I, t i ⊆ t N\n⊢ ∑ n ∈ I, μ (Td n) ≤ ⨆ n, μ (t n)... | calc
(∑ n ∈ I, μ (Td n)) = μ (⋃ n ∈ I, Td n) :=
(measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm
_ ≤ μ (⋃ n ∈ I, T n) := measure_mono (iUnion₂_mono fun n _hn => disjointed_subset _ _)
_ = μ (⋃ n ∈ I, t n) := measure_biUnion_toMeasurable I.countable... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 612,
"column": 4
} | {
"line": 612,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}\nT : Set (Set ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 13
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : μ s ≠ ∞\n⊢ μ.restrict (toMeasurable μ s) = μ.restrict s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 628,
"column": 43
} | {
"line": 628,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}\nT : Set (Set ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 385,
"column": 2
} | {
"line": 390,
"column": 96
} | [
{
"pp": "α : Type u_2\nι : Type u_6\nm0 : MeasurableSpace α\nμ ν : Measure α\ninst✝ : Countable ι\ns : ι → Set α\n⊢ μ.restrict (⋃ i, s i) = ν.restrict (⋃ i, s i) ↔ ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MeasureTheory.Measure",
"Chain... | refine ⟨fun h i => restrict_congr_mono (subset_iUnion _ _) h, fun h => ?_⟩
ext1 t ht
have D : Directed (· ⊆ ·) fun t : Finset ι => ⋃ i ∈ t, s i :=
Monotone.directed_le fun t₁ t₂ ht => biUnion_subset_biUnion_left ht
rw [iUnion_eq_iUnion_finset]
simp only [restrict_iUnion_apply_eq_iSup D ht, restrict_biUnion_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 385,
"column": 2
} | {
"line": 390,
"column": 96
} | [
{
"pp": "α : Type u_2\nι : Type u_6\nm0 : MeasurableSpace α\nμ ν : Measure α\ninst✝ : Countable ι\ns : ι → Set α\n⊢ μ.restrict (⋃ i, s i) = ν.restrict (⋃ i, s i) ↔ ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MeasureTheory.Measure",
"Chain... | refine ⟨fun h i => restrict_congr_mono (subset_iUnion _ _) h, fun h => ?_⟩
ext1 t ht
have D : Directed (· ⊆ ·) fun t : Finset ι => ⋃ i ∈ t, s i :=
Monotone.directed_le fun t₁ t₂ ht => biUnion_subset_biUnion_left ht
rw [iUnion_eq_iUnion_finset]
simp only [restrict_iUnion_apply_eq_iSup D ht, restrict_biUnion_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 569,
"column": 4
} | {
"line": 569,
"column": 19
} | [
{
"pp": "case h\nα : Type u_1\nι : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : Preorder ι\ninst✝¹ : IsCodirectedOrder ι\ninst✝ : atBot.IsCountablyGenerated\ns : ι → Set α\nhs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x\nhsm : ∀ (i : ι), NullMeasurableSet (s i) μ\ni✝ : ι\nhi : μ (s i✝) ≠ ∞\nthis : Non... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CountableSeparatingOn | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 37
} | [
{
"pp": "case h\nα : Type u_1\np : Set α → Prop\ns₀ : Set α\nhp : p s₀\nt : Set α\ninst✝ : HasCountableSeparatingOn α p t\nS : ℕ → Set α\nhSne : (range S).Nonempty\nhSc : (range S).Countable\nhS : (∀ s ∈ range S, p s) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ range S, x ∈ s ↔ y ∈ s) → x = y\n⊢ (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 537,
"column": 25
} | {
"line": 537,
"column": 55
} | [
{
"pp": "α : Type u_2\nι : Type u_6\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable ι\ns : ι → Set α\nt : Set α\nht : MeasurableSet t\n⊢ (μ.restrict (⋃ i, s i)) t ≤ (sum fun i ↦ μ.restrict (s i)) t",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 541,
"column": 25
} | {
"line": 541,
"column": 55
} | [
{
"pp": "α : Type u_2\nι : Type u_6\nm0 : MeasurableSpace α\nμ : Measure α\ns : ι → Set α\nT : Set ι\nhT : Countable ↑T\nt : Set α\nht : MeasurableSet t\n⊢ (μ.restrict (⋃ i ∈ T, s i)) t ≤ (sum fun i ↦ μ.restrict (s ↑i)) t",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 616,
"column": 2
} | {
"line": 616,
"column": 37
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\np : α → Prop\nh : (μ.restrict s) {a | ¬p a} = 0\n⊢ μ {a | ¬(a ∈ s → p a)} = 0",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"setOf",
"Membership.mem",
"id",
"Set.instInter... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 693,
"column": 4
} | {
"line": 693,
"column": 54
} | [
{
"pp": "case refine_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : Group β\nf g : α → β\nh : f / g =ᶠ[ae μ] 1\nx : α\nhx : (f / g) x = 1 x\n⊢ f x = g x",
"usedConstants": [
"instHDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
... | rwa [Pi.div_apply, Pi.one_apply, div_eq_one] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 693,
"column": 4
} | {
"line": 693,
"column": 54
} | [
{
"pp": "case refine_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : Group β\nf g : α → β\nh : f / g =ᶠ[ae μ] 1\nx : α\nhx : (f / g) x = 1 x\n⊢ f x = g x",
"usedConstants": [
"instHDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
... | rwa [Pi.div_apply, Pi.one_apply, div_eq_one] at hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 693,
"column": 4
} | {
"line": 693,
"column": 54
} | [
{
"pp": "case refine_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : Group β\nf g : α → β\nh : f / g =ᶠ[ae μ] 1\nx : α\nhx : (f / g) x = 1 x\n⊢ f x = g x",
"usedConstants": [
"instHDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
... | rwa [Pi.div_apply, Pi.one_apply, div_eq_one] at hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 763,
"column": 6
} | {
"line": 763,
"column": 70
} | [
{
"pp": "case h\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : NullMeasurableSet s μ\nt : Set α\nh : NullMeasurableSet t (μ.restrict s)\nt' : Set α\nht' : MeasurableSet t'\nt't : t' =ᶠ[ae (μ.restrict s)] t\nthis : ∀ᵐ (x : α) ∂μ, x ∈ s → (x ∈ t') = (x ∈ t)\ny : α\nhy : y ∈ s → (y ∈ t') = (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 775,
"column": 4
} | {
"line": 775,
"column": 15
} | [
{
"pp": "case refine_2\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : NullMeasurableSet s μ\nt : Set α\nh : NullMeasurableSet (t ∩ s) μ\nA : NullMeasurableSet (t \\ s) (μ.restrict s)\nB : NullMeasurableSet (t ∩ s) (μ.restrict s)\n⊢ NullMeasurableSet t (μ.restrict s)",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 784,
"column": 44
} | {
"line": 784,
"column": 55
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nht : t ⊆ s\nh : NullMeasurableSet t (μ.restrict s)\nt' : Set α\nt'_subs : t' ⊆ t\nht' : MeasurableSet t'\nt't : t' =ᶠ[ae (μ.restrict s)] t\nx : α\nhx : t' x = t x\n⊢ x ∈ t' ↔ x ∈ t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 805,
"column": 4
} | {
"line": 807,
"column": 42
} | [
{
"pp": "case hC\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nt : Set ↑s\nhs : NullMeasurableSet s μ\nt' : Set ↑s\nht' : t' ∈ {t | ∃ s_1, MeasurableSet s_1 ∧ Subtype.val ⁻¹' s_1 = t}\nht✝ : MeasurableSet t'\n⊢ NullMeasurableSet (Subtype.val '' t') μ",
"usedConstants": [
"Eq.mpr",
... | obtain ⟨s', hs', rfl⟩ := ht'
rw [Subtype.image_preimage_coe]
exact hs.inter (hs'.nullMeasurableSet) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 805,
"column": 4
} | {
"line": 807,
"column": 42
} | [
{
"pp": "case hC\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nt : Set ↑s\nhs : NullMeasurableSet s μ\nt' : Set ↑s\nht' : t' ∈ {t | ∃ s_1, MeasurableSet s_1 ∧ Subtype.val ⁻¹' s_1 = t}\nht✝ : MeasurableSet t'\n⊢ NullMeasurableSet (Subtype.val '' t') μ",
"usedConstants": [
"Eq.mpr",
... | obtain ⟨s', hs', rfl⟩ := ht'
rw [Subtype.image_preimage_coe]
exact hs.inter (hs'.nullMeasurableSet) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\ns : Set α\nhs : MeasurableSet s\n⊢ s = ⋃ y ∈ s, measurableAtom y",
"usedConstants": [
"Set.Subset.antisymm",
"Membership.mem",
"measurableAtom",
"Set.instMembership",
"Set.iUnion",
"Set"... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 121,
"column": 8
} | {
"line": 121,
"column": 19
} | [
{
"pp": "case h₁\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\ns : Set α\nhs : MeasurableSet s\nx : α\nhx : x ∈ s\n⊢ x ∈ ⋃ y ∈ s, measurableAtom y",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membership.mem",
"Exists",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 174,
"column": 26
} | {
"line": 185,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace β\ninst✝¹ : Nonempty β\ninst✝ : SeparatesPoints β\nf : α → β\nhf : Measurable f\n⊢ ∃ c, f = fun x ↦ c",
"usedConstants": [
"Classical.ofNonempty",
"False",
"Lattice.toSemilatticeSup",
"MeasurableSet",
"Set.mem_empty_... | by
have h (a₁ : α) (a₂ : α) : f a₁ = f a₂ := by
by_contra! h
obtain ⟨s, hs, hx, hy⟩ := exists_measurableSet_of_ne h
obtain h' | h' := MeasurableSpace.measurableSet_bot_iff.mp (hf hs)
· absurd hx
simp [← mem_preimage, h']
· absurd hy
simp [← mem_preimage, h']
obtain h' | h' := isEmpty... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 592,
"column": 42
} | {
"line": 592,
"column": 65
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable α\nhμ : ∀ (a : α), μ {a} < ∞\nh✝ : Nonempty α\nf : ℕ → α\nhf : Surjective f\n⊢ ∀ (i : ℕ), μ ((fun n ↦ {f n}) i) < ∞",
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"PartialOrder.toPreorder",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 606,
"column": 38
} | {
"line": 606,
"column": 62
} | [
{
"pp": "α : Type u_1\nμ : Measure α\nh this : SigmaFinite μ\ns : ℕ → Set α := spanningSets μ\nhs_univ : ⋃ i, s i = univ\nhs_meas : ∀ (i : ℕ), s i = ∅ ∨ s i = univ\nh_univ_empty : ¬univ = ∅\nh_not_univ : ∀ (i : ℕ), s i ≠ univ\n⊢ ∀ (i : ℕ), s i = ∅",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.MeasurePreserving | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf : α → β\nhf : MeasurePreserving f μa μb\nh₂ : MeasurableEmbedding f\ns : Set α\n⊢ MeasurePreserving f (μa.restrict s) (μb.restrict (f '' s))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1074,
"column": 4
} | {
"line": 1076,
"column": 16
} | [
{
"pp": "case h\nα : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → β\ninst✝ : Zero β\ng : α → β\nhs : MeasurableSet s\nh : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x\nx : α\nhx : x ∈ s → f x = g x\n⊢ s.indicator f x = s.indicator g x",
"usedConstants": [
"False",
"e... | by_cases hxs : x ∈ s
· simp [hxs, hx hxs]
· simp [hxs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1074,
"column": 4
} | {
"line": 1076,
"column": 16
} | [
{
"pp": "case h\nα : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → β\ninst✝ : Zero β\ng : α → β\nhs : MeasurableSet s\nh : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x\nx : α\nhx : x ∈ s → f x = g x\n⊢ s.indicator f x = s.indicator g x",
"usedConstants": [
"False",
"e... | by_cases hxs : x ∈ s
· simp [hxs, hx hxs]
· simp [hxs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1078,
"column": 4
} | {
"line": 1078,
"column": 21
} | [
{
"pp": "case h\nα : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → β\ninst✝ : Zero β\ng : α → β\nhs : MeasurableSet s\nh : s.indicator f =ᶠ[ae μ] s.indicator g\nx : α\nhx : s.indicator f x = s.indicator g x\nhxs : x ∈ s\n⊢ f x = g x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.MeasurePreserving | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → α\ns : Set α\nhf : MeasurePreserving f μ μ\nhs : NullMeasurableSet s μ\nn : ℕ\nhvol : μ univ < ↑n * μ s\nA : ∀ (m : ℕ), NullMeasurableSet (f^[m] ⁻¹' s) μ\nB : ∀ (m : ℕ), μ (f^[m] ⁻¹' s) = μ s\nthis : μ univ < ∑ m ∈ Finset.range n, μ (f^[m]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 211,
"column": 6
} | {
"line": 211,
"column": 25
} | [
{
"pp": "case pos\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nμ : Measure α\nH : AEMeasurable f μ\nt : Set β\nht : ∀ᵐ (x : α) ∂μ, f x ∈ t\nh₀ : t.Nonempty\ns : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ\ng : α → β := s.piecewise (fun x ↦ h₀.some) (mk f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 62
} | [
{
"pp": "case neg\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nμ : Measure α\nH : AEMeasurable f μ\nt : Set β\nht : ∀ᵐ (x : α) ∂μ, f x ∈ t\nh₀ : t.Nonempty\ns : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ\ng : α → β := s.piecewise (fun x ↦ h₀.some) (mk f... | simp only [g, hx, piecewise_eq_of_notMem, not_false_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1164,
"column": 4
} | {
"line": 1201,
"column": 40
} | [
{
"pp": "case refine_2\nα : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nt' : ℕ → Set α\nht' : s ⊆ iUnion t'\n⊢ sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)} ≤ ∑' (n : ℕ), ⨅ μ_1 ∈ toOuterMeasure '' {μ, ν}, μ_1 (t' n)",
"usedConstants": [
"Iff.mpr",
"Set.ext",
... | simp only [iInf_image, coe_toOuterMeasure, iInf_pair]
-- Conversely, fixing `t' : ℕ → Set α` such that `s ⊆ ⋃ n, t' n`, we construct `t : Set α`
-- for which `μ (t ∩ s) + ν (tᶜ ∩ s) ≤ ∑' n, μ (t' n) ⊓ ν (t' n)`.
-- Denoting `I := {n | μ (t' n) ≤ ν (t' n)}`, we set `t = ⋃ n ∈ I, t' n`.
-- Clearly `μ (t ∩... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 384,
"column": 25
} | {
"line": 384,
"column": 64
} | [
{
"pp": "α : Type u_2\nδ : Type u_5\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace δ\nμ ν : Measure α\nf : α → δ\nh : μ ≤ ν\nhf : AEMeasurable f ν\ns : Set δ\nhs : MeasurableSet s\n⊢ (map f μ) s ≤ (map f ν) s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1164,
"column": 4
} | {
"line": 1201,
"column": 40
} | [
{
"pp": "case refine_2\nα : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nt' : ℕ → Set α\nht' : s ⊆ iUnion t'\n⊢ sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)} ≤ ∑' (n : ℕ), ⨅ μ_1 ∈ toOuterMeasure '' {μ, ν}, μ_1 (t' n)",
"usedConstants": [
"Iff.mpr",
"Set.ext",
... | simp only [iInf_image, coe_toOuterMeasure, iInf_pair]
-- Conversely, fixing `t' : ℕ → Set α` such that `s ⊆ ⋃ n, t' n`, we construct `t : Set α`
-- for which `μ (t ∩ s) + ν (tᶜ ∩ s) ≤ ∑' n, μ (t' n) ⊓ ν (t' n)`.
-- Denoting `I := {n | μ (t' n) ≤ ν (t' n)}`, we set `t = ⋃ n ∈ I, t' n`.
-- Clearly `μ (t ∩... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1244,
"column": 4
} | {
"line": 1244,
"column": 37
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : NeZero μ\nh : IsEmpty α\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1295,
"column": 25
} | {
"line": 1295,
"column": 58
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm0 : MeasurableSpace α\nμ : ι → Measure α\ni : ι\ns : Set α\nhs : MeasurableSet s\n⊢ (μ i) s ≤ (sum μ) s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
"MeasureTheory.Measure.sum_apply",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1371,
"column": 2
} | {
"line": 1371,
"column": 29
} | [
{
"pp": "case h\nα : 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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Equiv.instEquivLike",
"ENNReal.instAddCommMonoid... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 420,
"column": 29
} | {
"line": 420,
"column": 40
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\nf : α → β\nh : NullMeasurable f μ\ninst✝ : MeasurableSpace.CountablyGenerated ↑∅\nhft : ∀ᵐ (x : α) ∂μ, f x ∈ ∅\n⊢ μ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.GDelta | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 60
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ns t : Set X\ninst✝ : NormalSpace X\nst_dis : Disjoint s t\nt_cl : IsClosed[inst✝¹] t\nT : Set (Set X)\nT_open : ∀ t ∈ T, IsOpen[inst✝¹] t\nT_count : T.Countable\nT_int : t = ⋂₀ T\nT_nonempty : T.Nonempty\n⊢ HasSeparatingCover s t",
"usedConstants... | obtain ⟨g, g_surj⟩ := T_count.exists_surjective T_nonempty | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 173,
"column": 6
} | {
"line": 177,
"column": 34
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nn : ℕ\n⊢ Measurable fun x ↦ (x, n).1 ^ (x, n).2",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"Measurable.mul",
"HMul.hMul",
"Monoid.toMulOneClass"... | induction n with
| zero => simp only [pow_zero, ← Pi.one_def, measurable_one]
| succ n ih =>
simp only [pow_succ]
exact ih.mul measurable_id | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 173,
"column": 6
} | {
"line": 177,
"column": 34
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nn : ℕ\n⊢ Measurable fun x ↦ (x, n).1 ^ (x, n).2",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"Measurable.mul",
"HMul.hMul",
"Monoid.toMulOneClass"... | induction n with
| zero => simp only [pow_zero, ← Pi.one_def, measurable_one]
| succ n ih =>
simp only [pow_succ]
exact ih.mul measurable_id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 173,
"column": 6
} | {
"line": 177,
"column": 34
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nn : ℕ\n⊢ Measurable fun x ↦ (x, n).1 ^ (x, n).2",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"Measurable.mul",
"HMul.hMul",
"Monoid.toMulOneClass"... | induction n with
| zero => simp only [pow_zero, ← Pi.one_def, measurable_one]
| succ n ih =>
simp only [pow_succ]
exact ih.mul measurable_id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 363,
"column": 15
} | {
"line": 363,
"column": 41
} | [
{
"pp": "α : Type u_3\nm : MeasurableSpace α\nG : Type u_4\ninst✝² : InvolutiveInv G\ninst✝¹ : MeasurableSpace G\ninst✝ : MeasurableInv G\nf : α → G\nh : Measurable fun x ↦ (f x)⁻¹\n⊢ Measurable f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 368,
"column": 15
} | {
"line": 368,
"column": 41
} | [
{
"pp": "α : Type u_3\nm : MeasurableSpace α\nμ : Measure α\nG : Type u_4\ninst✝² : InvolutiveInv G\ninst✝¹ : MeasurableSpace G\ninst✝ : MeasurableInv G\nf : α → G\nh : AEMeasurable (fun x ↦ (f x)⁻¹) μ\n⊢ AEMeasurable f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 432,
"column": 23
} | {
"line": 432,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Group α\ninst✝ : MeasurableDiv α\n⊢ Measurable Inv.inv",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 490,
"column": 32
} | {
"line": 490,
"column": 48
} | [
{
"pp": "α✝ : Type u_1\nM : Type u_2\nα : Type u_3\ninst✝³ : MeasurableSpace α\ninst✝² : Monoid M\ninst✝¹ : MulAction M α\ninst✝ : MeasurableConstSMul M α\ns : Submonoid M\nc : ↥s\n⊢ Measurable fun x ↦ c • x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 532,
"column": 30
} | {
"line": 532,
"column": 41
} | [
{
"pp": "α✝ : Type u_1\nM : Type u_2\nX : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : SMul M X\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → M\ng : α → X\ninst✝⁴ : MeasurableConstSMul M X\ninst✝³ : SMul M α\ninst✝² : SMul Mᵐᵒᵖ α\ninst✝¹ : IsCentralScalar M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.