module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.SetDissipate | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 23
} | {
"line": 65,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : α → Set β\ninst✝ : Preorder α\n⊢ ⋂ x, dissipate s x = ⋂ x, s x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Set.dissipate",
"Set.iInter",
"Preorder.toLE"
],
"usedFVars": [
"β",
"... | [
"case h₁\nα : Type u_1\nβ : Type u_2\ns : α → Set β\ninst✝ : Preorder α\n⊢ ⋂ x, dissipate s x ⊆ ⋂ x, s x",
"case h₂\nα : Type u_1\nβ : Type u_2\ns : α → Set β\ninst✝ : Preorder α\n⊢ ⋂ x, s x ⊆ ⋂ x, dissipate s x"
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.PiSystem | {
"line": 117,
"column": 12
} | {
"line": 117,
"column": 23
} | {
"line": 117,
"column": 24
} | [
{
"pp": "case zero\nα : Type u_1\ns : ℕ → Set α\nC : Set (Set α)\nhC : IsPiSystem C\nh : ∀ (n : ℕ), s n ∈ C\nh' : (dissipate s 0).Nonempty\n⊢ dissipate s 0 ∈ C",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Set.dissipate",
"Eq.mpr",
"congrArg",
"Membership.mem",
... | [
"case zero\nα : Type u_1\ns : ℕ → Set α\nC : Set (Set α)\nhC : IsPiSystem C\nh : ∀ (n : ℕ), s n ∈ C\nh' : (dissipate s 0).Nonempty\n⊢ s 0 ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 646,
"column": 4
} | {
"line": 646,
"column": 37
} | {
"line": 646,
"column": 38
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ns : Set α\nx : α\nι : Sort u_4\nδ' : Type u_6\ninst✝ : ConditionallyCompleteLinearOrder δ'\nf : ι → α → δ'\nbdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i ↦ f i y)\nh : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x\nh✝ : IsEmpty ι\n⊢ LowerSemi... | [
"case inl\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ns : Set α\nx : α\nι : Sort u_4\nδ' : Type u_6\ninst✝ : ConditionallyCompleteLinearOrder δ'\nf : ι → α → δ'\nbdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i ↦ f i y)\nh : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x\nh✝ : IsEmpty ι\n⊢ LowerSemicontinuousWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 149,
"column": 19
} | {
"line": 149,
"column": 30
} | {
"line": 149,
"column": 31
} | [
{
"pp": "case singleton\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\nt : Finset (Set α)\na : Set α\nht : ∀ s ∈ {a}, s ∈ S\nh' : (⋂ s ∈ {a}, s).Nonempty\n⊢ ⋂ s ∈ {a}, s ∈ S",
"ppTerm": "?singleton",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Iff.of_eq",
"congrArg",
... | [
"case singleton\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\nt : Finset (Set α)\na : Set α\nht : ∀ s ∈ {a}, s ∈ S\nh' : (⋂ s ∈ {a}, s).Nonempty\n⊢ a ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 37
} | {
"line": 198,
"column": 38
} | [
{
"pp": "α : Type u_1\nι : Sort u_3\nι' : Sort u_4\ninst✝ : LinearOrder α\nIxx : α → α → Set α\np : α → α → Prop\nHne : ∀ {a b : α}, (Ixx a b).Nonempty → p a b\nHi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)\nf : ι → α\ng : ι' → α\n⊢ IsPiSystem {S | ∃ i j, p (f i) (g j) ∧ Ixx (f i... | [
"α : Type u_1\nι : Sort u_3\nι' : Sort u_4\ninst✝ : LinearOrder α\nIxx : α → α → Set α\np : α → α → Prop\nHne : ∀ {a b : α}, (Ixx a b).Nonempty → p a b\nHi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)\nf : ι → α\ng : ι' → α\n⊢ IsPiSystem {S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 684,
"column": 78
} | {
"line": 684,
"column": 95
} | {
"line": 685,
"column": 8
} | [
{
"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",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Real",
"DivI... | [
"k : ℕ\nhn : 0 < k.succ\n⊢ ↑(∏ i ∈ Finset.range k.succ, (i + 1)) * (∏ _k ∈ Finset.range k.succ, ↑k.succ)⁻¹ ≤ (↑k.succ)⁻¹"
] | ← inv_eq_one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 15
} | {
"line": 291,
"column": 16
} | [
{
"pp": "case base\nα : Type u_3\nβ : Type u_4\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nb : β\nh_s_in_t' : s ∈ (fun b ↦ g b) b\n⊢ s = ⋂ b_1 ∈ {b}, (fun x ↦ s) b_1 ∧ ∀ b_1 ∈ {b}, (fun x ↦ s) b_1 ∈ g b_1",
"ppTerm": "?base",
"assigned": true,
"usedConstants": [
"Eq.... | [
"case base\nα : Type u_3\nβ : Type u_4\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nb : β\nh_s_in_t' : s ∈ (fun b ↦ g b) b\n⊢ s ∈ g b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 740,
"column": 2
} | {
"line": 740,
"column": 13
} | {
"line": 740,
"column": 14
} | [
{
"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)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 588,
"column": 2
} | {
"line": 588,
"column": 29
} | {
"line": 588,
"column": 30
} | [
{
"pp": "α : Type u_2\nf : Filter α\nm : α → EReal\na b : EReal\nhm : Tendsto m f (𝓝 a)\nh₁ : a ≠ 0 ∨ b ≠ ⊥\nh₂ : a ≠ 0 ∨ b ≠ ⊤\n⊢ Tendsto (fun x ↦ m x * b) f (𝓝 (a * b))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"HMul.... | [
"α : Type u_2\nf : Filter α\nm : α → EReal\na b : EReal\nhm : Tendsto m f (𝓝 a)\nh₁ : a ≠ 0 ∨ b ≠ ⊥\nh₂ : a ≠ 0 ∨ b ≠ ⊤\n⊢ Tendsto (fun x ↦ b * m x) f (𝓝 (a * b))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 70,
"column": 15
} | {
"line": 70,
"column": 26
} | {
"line": 70,
"column": 27
} | [
{
"pp": "α : Type u\nm : OuterMeasure α\ns : Set α\nh : m.IsCaratheodory sᶜ\n⊢ m.IsCaratheodory s",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nm : OuterMeasure α\ns : Set α\nh : m.IsCaratheodory sᶜ\n⊢ m.IsCaratheodory s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 25
} | {
"line": 119,
"column": 26
} | [
{
"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)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.to... | [
"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 (⋃ x ∈ Iic i, s x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 756,
"column": 2
} | {
"line": 756,
"column": 13
} | {
"line": 756,
"column": 14
} | [
{
"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)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 15
} | {
"line": 134,
"column": 16
} | [
{
"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 ∈ ∅",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
... | [
"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 → ∀ x < n, a ∉ s x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.AE | {
"line": 128,
"column": 26
} | {
"line": 128,
"column": 42
} | {
"line": 128,
"column": 43
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 33
} | {
"line": 350,
"column": 34
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[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",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],... | [
"case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 537,
"column": 15
} | {
"line": 537,
"column": 26
} | {
"line": 537,
"column": 27
} | [
{
"pp": "α : Type u_3\nd : DynkinSystem α\na : Set α\nh : d.Has aᶜ\n⊢ d.Has a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_3\nd : DynkinSystem α\na : Set α\nh : d.Has aᶜ\n⊢ d.Has a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 539,
"column": 36
} | {
"line": 539,
"column": 47
} | {
"line": 539,
"column": 48
} | [
{
"pp": "α : Type u_3\nd : DynkinSystem α\n⊢ d.Has univ",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_3\nd : DynkinSystem α\n⊢ d.Has univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Induced | {
"line": 387,
"column": 58
} | {
"line": 387,
"column": 87
} | {
"line": 387,
"column": 88
} | [
{
"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",
"ppTerm": "?m.100",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 35
} | {
"line": 310,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\n⊢ MeasurableSet (limsup s atTop)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"CompleteLattice.toConditionallyCompleteLatt... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\n⊢ MeasurableSet (blimsup s atTop fun x ↦ True)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 35
} | {
"line": 315,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\n⊢ MeasurableSet (liminf s atTop)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"_private.Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGene... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\n⊢ MeasurableSet (bliminf s atTop fun x ↦ True)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 76,
"column": 15
} | {
"line": 76,
"column": 61
} | {
"line": 76,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nmα : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\nh : MeasurableSet (f '' s)\n⊢ MeasurableSet s",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nmα : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\nh : MeasurableSet (f '' s)\n⊢ MeasurableSet s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 332,
"column": 15
} | {
"line": 332,
"column": 56
} | {
"line": 332,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ne : α ≃ᵐ β\ns : Set β\nh : MeasurableSet (⇑e ⁻¹' s)\n⊢ MeasurableSet s",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ne : α ≃ᵐ β\ns : Set β\nh : MeasurableSet (⇑e ⁻¹' s)\n⊢ MeasurableSet s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AEDisjoint | {
"line": 139,
"column": 6
} | {
"line": 139,
"column": 17
} | {
"line": 139,
"column": 18
} | [
{
"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",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"id",
"SDiff.sdiff",
"And",
... | [
"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 ∧ x ∉ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Monotone | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 51
} | {
"line": 210,
"column": 4
} | [
{
"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\nih : k ≤ φ k\nhφ : StrictMonoOn φ (Iic (succ k))\nhk : ¬IsMax k\nhm : succ k ≤ succ k\n⊢ succ (φ k) ≤ φ (succ k)",
"ppTerm": "?neg.inl✝",
"assigne... | [
"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\nih : k ≤ φ k\nhφ : StrictMonoOn φ (Iic (succ k))\nhk : ¬IsMax k\nhm : succ k ≤ succ k\n⊢ k < succ k"
] | refine succ_le_of_lt (hφ (le_succ _) le_rfl ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 668,
"column": 8
} | {
"line": 668,
"column": 56
} | {
"line": 668,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (range f)\ns : Set β\nhs : MeasurableSet s\n⊢ MeasurableSet (f '' f ⁻¹' s)",
"ppTerm": "?m.88",
"assigned": true,
"usedCons... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (range f)\ns : Set β\nhs : MeasurableSet s\n⊢ MeasurableSet (s ∩ range f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 677,
"column": 4
} | {
"line": 678,
"column": 11
} | {
"line": 678,
"column": 12
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u✝ : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → α\ns : Set α\nhf : MeasurableEmbedding f\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (⇑(Equiv.... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u✝ : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → α\ns : Set α\nhf : MeasurableEmbedding f\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (Subtype.val ⁻¹' f ''... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 34
} | {
"line": 205,
"column": 35
} | [
{
"pp": "case refine_2\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhtm : MeasurableSet t\nhst : s =ᵐ[μ] t\nthis : toMeasurable μ (s \\ t) =ᵐ[μ] ∅\n⊢ t ∪ toMeasurable μ (s \\ t) =ᵐ[μ] s",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"case refine_2\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhtm : MeasurableSet t\nhst : s =ᵐ[μ] t\nthis : toMeasurable μ (s \\ t) =ᵐ[μ] ∅\n⊢ t ∪ toMeasurable μ (s \\ t) =ᵐ[μ] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 18
} | {
"line": 309,
"column": 19
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : μ sᶜ = 0\n⊢ μ s = μ univ",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : μ sᶜ = 0\n⊢ μ s = μ univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Map | {
"line": 123,
"column": 38
} | {
"line": 123,
"column": 74
} | {
"line": 123,
"column": 75
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf g : α → β\nh : f =ᵐ[μ] g\nhf : ¬AEMeasurable f μ\n⊢ ¬AEMeasurable g μ",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AEMeasurable",
"congrArg",
"id",
... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf g : α → β\nh : f =ᵐ[μ] g\nhf : ¬AEMeasurable f μ\n⊢ ¬AEMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Map | {
"line": 129,
"column": 54
} | {
"line": 129,
"column": 65
} | {
"line": 129,
"column": 66
} | [
{
"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",
"ppTerm": "?m.43",
"assigned": fa... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | {
"line": 89,
"column": 41
} | {
"line": 89,
"column": 61
} | {
"line": 89,
"column": 62
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Measure... | [
"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⊢ ν (f ⁻¹' s) = 0 → μ (f ⁻¹' s) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | {
"line": 112,
"column": 65
} | {
"line": 112,
"column": 76
} | {
"line": 112,
"column": 77
} | [
{
"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",
"ppTerm": "?m.62",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.BorelCantelli | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 57
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case e_6\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",
"ppTerm": "?e_6",
"assigned": true,
... | [] | simp [mem_limsup_iff_frequently_mem, Filter.Frequently] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Map | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 30
} | {
"line": 316,
"column": 31
} | [
{
"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",
"ppTerm": "?m.87",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 46
} | {
"line": 164,
"column": 47
} | [
{
"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⊢ (preimage f)^[n] s =ᵐ[μ] s",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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⊢ (preimage f)^[n] s =ᵐ[μ] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 46
} | {
"line": 170,
"column": 47
} | [
{
"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⊢ (preimage f)^[n] s =ᵐ[μ] s",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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⊢ (preimage f)^[n] s =ᵐ[μ] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 55
} | {
"line": 171,
"column": 56
} | [
{
"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 ... | [
"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) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 57
} | {
"line": 174,
"column": 58
} | [
{
"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))",
"p... | [
"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 → ∃ x, MeasurableSet x ∧ ¬NullMeasurableSet (f '' x) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 9
} | {
"line": 148,
"column": 10
} | [
{
"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)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"MeasureTheory.Measure",
"Lattice.toS... | [
"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 ∪ t \\ s) = μ (s \\ t) + μ (t \\ s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 45,
"column": 6
} | {
"line": 45,
"column": 17
} | {
"line": 45,
"column": 18
} | [
{
"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 < ∞",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Measure... | [
"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 s < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 54
} | {
"line": 149,
"column": 55
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nhμ : measureUnivNNReal μ ≤ 0\n⊢ μ = 0",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nhμ : measureUnivNNReal μ ≤ 0\n⊢ μ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 351,
"column": 17
} | {
"line": 351,
"column": 52
} | {
"line": 351,
"column": 53
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh : s ∪ t =ᵐ[μ] s\n⊢ μ (t \\ s) = 0",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh : s ∪ t =ᵐ[μ] s\n⊢ μ (t \\ s) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 69,
"column": 18
} | {
"line": 69,
"column": 29
} | {
"line": 69,
"column": 30
} | [
{
"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... | [
"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⟩ ≤ ⟨c, hcs⟩... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 60
} | {
"line": 364,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh₁ : s ≤ᵐ[μ] t\nh₂ : μ t ≤ μ s\nhsm : NullMeasurableSet s μ\nht : μ t ≠ ∞\n⊢ μ (t \\ s) = 0",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"LE.le.antisymm",
"MeasureTheory.... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh₁ : s ≤ᵐ[μ] t\nhsm : NullMeasurableSet s μ\nht : μ t ≠ ∞\nh₂ : μ t = μ s\n⊢ μ (t \\ s) = 0"
] | replace h₂ : μ t = μ s := h₂.antisymm (measure_mono_ae h₁) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 70,
"column": 18
} | {
"line": 70,
"column": 29
} | {
"line": 70,
"column": 30
} | [
{
"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... | [
"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⟩ ≤ ⟨c, hcs⟩... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 71,
"column": 18
} | {
"line": 71,
"column": 29
} | {
"line": 71,
"column": 30
} | [
{
"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... | [
"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⟩ ≤ ⟨c, hcs⟩... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 30
} | {
"line": 306,
"column": 31
} | [
{
"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 < ∞",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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 < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 32
} | {
"line": 111,
"column": 33
} | [
{
"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",
"ppTerm": "?m.47",
"... | [
"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, a < a_1 ∧ Ioo a a_1 ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 44
} | {
"line": 130,
"column": 45
} | [
{
"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",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"M... | [
"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.restrict t) s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 39
} | {
"line": 547,
"column": 40
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"MeasureTheory.ae",
"Eq.mpr",
... | [
"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, tᶜ ∈ ae μ) → sᶜ ∈ ae μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 508,
"column": 6
} | {
"line": 514,
"column": 45
} | {
"line": 516,
"column": 0
} | [
{
"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.SFinite | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | {
"line": 74,
"column": 14
} | [
{
"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)",
"ppTerm": "?m.18",
"assigned": false,
"usedConsta... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 53
} | {
"line": 271,
"column": 54
} | [
{
"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",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 46
} | {
"line": 297,
"column": 47
} | [
{
"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",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"MeasureTheory.Measure",
"co... | [
"α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns s' t : Set α\nht : MeasurableSet t\n⊢ μ (t ∩ s ∪ t ∩ s') ≤ μ (t ∩ s) + μ (t ∩ s')"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 24
} | {
"line": 92,
"column": 25
} | [
{
"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 < ∞",
"ppTerm": "?m... | [
"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⊢ ∑' (i : ℕ), ↑(c i) * (sfiniteSeq μ i) univ < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 13
} | {
"line": 333,
"column": 14
} | [
{
"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",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 604,
"column": 4
} | {
"line": 604,
"column": 19
} | {
"line": 604,
"column": 20
} | [
{
"pp": "α : 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 : Nonempty ι\... | [
"α : 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 : Nonempty ι\nt : ι → Set... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 387,
"column": 2
} | {
"line": 392,
"column": 96
} | {
"line": 394,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr"... | [] | 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": 387,
"column": 2
} | {
"line": 392,
"column": 96
} | {
"line": 394,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr"... | [] | 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.Order.Filter.CountableSeparatingOn | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 37
} | {
"line": 111,
"column": 38
} | [
{
"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, ... | [
"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, ∀ y ∈ t, (∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 539,
"column": 25
} | {
"line": 539,
"column": 55
} | {
"line": 539,
"column": 56
} | [
{
"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",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheo... | [
"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 (⋃ i, t ∩ s i) ≤ ∑' (i : ι), μ (t ∩ s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 543,
"column": 25
} | {
"line": 543,
"column": 55
} | {
"line": 543,
"column": 56
} | [
{
"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",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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 (⋃ i ∈ T, t ∩ s i) ≤ ∑' (i : ↑T), μ (t ∩ s ↑i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 37
} | {
"line": 618,
"column": 38
} | [
{
"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",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"setOf",
"Memb... | [
"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 (s ∩ {a | ¬p a}) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 695,
"column": 4
} | {
"line": 695,
"column": 54
} | {
"line": 696,
"column": 2
} | [
{
"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 =ᵐ[μ] 1\nx : α\nhx : (f / g) x = 1 x\n⊢ f x = g x",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"instHDiv",
"InvOneClass.toOne",
"DivIn... | [] | 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": 695,
"column": 4
} | {
"line": 695,
"column": 54
} | {
"line": 696,
"column": 2
} | [
{
"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 =ᵐ[μ] 1\nx : α\nhx : (f / g) x = 1 x\n⊢ f x = g x",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"instHDiv",
"InvOneClass.toOne",
"DivIn... | [] | 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": 695,
"column": 4
} | {
"line": 695,
"column": 54
} | {
"line": 696,
"column": 2
} | [
{
"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 =ᵐ[μ] 1\nx : α\nhx : (f / g) x = 1 x\n⊢ f x = g x",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"instHDiv",
"InvOneClass.toOne",
"DivIn... | [] | 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": 765,
"column": 6
} | {
"line": 765,
"column": 70
} | {
"line": 765,
"column": 71
} | [
{
"pp": "α : 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' =ᵐ[μ.restrict s] t\nthis : ∀ᵐ (x : α) ∂μ, x ∈ s → (x ∈ t') = (x ∈ t)\ny : α\nhy : y ∈ s → (y ∈ t') = (y ∈ t)\n⊢ (y ... | [
"α : 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' =ᵐ[μ.restrict s] t\nthis : ∀ᵐ (x : α) ∂μ, x ∈ s → (x ∈ t') = (x ∈ t)\ny : α\nhy : y ∈ s → (y ∈ t') = (y ∈ t)\n⊢ y ∈ s → (y ∈ t'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 27
} | {
"line": 120,
"column": 6
} | [
{
"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",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Membership.mem",
"measurableAtom",
"Set.ins... | [
"case h₁\nα : 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",
"case h₂\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\ns : Set α\nhs : MeasurableSet s\n⊢ ⋃ y ∈ s, measurableAtom y ⊆ s"
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 121,
"column": 8
} | {
"line": 121,
"column": 19
} | {
"line": 121,
"column": 20
} | [
{
"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",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.mem_iUnion._simp_1",... | [
"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⊢ ∃ i ∈ s, x ∈ measurableAtom i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 777,
"column": 4
} | {
"line": 777,
"column": 15
} | {
"line": 777,
"column": 16
} | [
{
"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)",
"ppTerm": "?refi... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 194,
"column": 6
} | {
"line": 194,
"column": 23
} | {
"line": 194,
"column": 24
} | [
{
"pp": "case refine_1.refine_2\nα : Type u_1\nmα : MeasurableSpace α\ninst✝ : CountablyGenerated α\ns : Set α\nhs : MeasurableSet s\nn : ℕ\nx✝ : MeasurableSet (natGeneratingSequence α n)\nx : α\np : ℕ → Prop\nhpn : p n\nhpx : x ∈ if p n then natGeneratingSequence α n else (natGeneratingSequence α n)ᶜ\n⊢ x ∈ na... | [
"case refine_1.refine_2\nα : Type u_1\nmα : MeasurableSpace α\ninst✝ : CountablyGenerated α\ns : Set α\nhs : MeasurableSet s\nn : ℕ\nx✝ : MeasurableSet (natGeneratingSequence α n)\nx : α\np : ℕ → Prop\nhpn : p n\nhpx : x ∈ if p n then natGeneratingSequence α n else (natGeneratingSequence α n)ᶜ\n⊢ x ∈ natGeneratingS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 807,
"column": 4
} | {
"line": 809,
"column": 42
} | {
"line": 810,
"column": 2
} | [
{
"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",
"ppTerm": "?hC",
"assigned": tru... | [] | 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": 807,
"column": 4
} | {
"line": 809,
"column": 42
} | {
"line": 810,
"column": 2
} | [
{
"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",
"ppTerm": "?hC",
"assigned": tru... | [] | 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": 296,
"column": 26
} | {
"line": 307,
"column": 60
} | {
"line": 309,
"column": 0
} | [
{
"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",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Classical.ofNonempty",
"False",
"Lattice.toSemilatticeSup",... | [] | 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.Restrict | {
"line": 1075,
"column": 4
} | {
"line": 1077,
"column": 16
} | {
"line": 1078,
"column": 2
} | [
{
"pp": "α : 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",
"ppTerm": "?m.59",
"assigned": true,
"used... | [] | by_cases hxs : x ∈ s
· simp [hxs, hx hxs]
· simp [hxs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1075,
"column": 4
} | {
"line": 1077,
"column": 16
} | {
"line": 1078,
"column": 2
} | [
{
"pp": "α : 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",
"ppTerm": "?m.59",
"assigned": true,
"used... | [] | by_cases hxs : x ∈ s
· simp [hxs, hx hxs]
· simp [hxs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1079,
"column": 4
} | {
"line": 1079,
"column": 21
} | {
"line": 1079,
"column": 22
} | [
{
"pp": "α : 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 =ᵐ[μ] s.indicator g\nx : α\nhx : s.indicator f x = s.indicator g x\nhxs : x ∈ s\n⊢ f x = g x",
"ppTerm": "?m.89",
"assigned": false,
... | [
"α : 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 =ᵐ[μ] s.indicator g\nx : α\nhx : s.indicator f x = s.indicator g x\nhxs : x ∈ s\n⊢ f x = g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 593,
"column": 42
} | {
"line": 593,
"column": 65
} | {
"line": 593,
"column": 66
} | [
{
"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 {f i} < ∞",
"ppTerm": "?m.92",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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 {f i} < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 607,
"column": 38
} | {
"line": 607,
"column": 62
} | {
"line": 607,
"column": 63
} | [
{
"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 = ∅",
"ppTerm": "?m.104",
"assigned": false,
"usedConsta... | [
"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 = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 211,
"column": 6
} | {
"line": 211,
"column": 25
} | {
"line": 211,
"column": 26
} | [
{
"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... | [
"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 H)\nx : α\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 62
} | {
"line": 213,
"column": 6
} | [
{
"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... | [
"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 H)\nx : α\n... | simp only [g, hx, piecewise_eq_of_notMem, not_false_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 384,
"column": 25
} | {
"line": 384,
"column": 64
} | {
"line": 384,
"column": 65
} | [
{
"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",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureT... | [
"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⊢ μ (f ⁻¹' s) ≤ ν (f ⁻¹' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 171,
"column": 6
} | {
"line": 175,
"column": 34
} | {
"line": 175,
"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",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"Measurable.mul",
... | [] | 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": 171,
"column": 6
} | {
"line": 175,
"column": 34
} | {
"line": 175,
"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",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"Measurable.mul",
... | [] | 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": 171,
"column": 6
} | {
"line": 175,
"column": 34
} | {
"line": 175,
"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",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"Measurable.mul",
... | [] | 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": 359,
"column": 15
} | {
"line": 359,
"column": 41
} | {
"line": 359,
"column": 42
} | [
{
"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",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 364,
"column": 15
} | {
"line": 364,
"column": 41
} | {
"line": 364,
"column": 42
} | [
{
"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",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 428,
"column": 23
} | {
"line": 428,
"column": 34
} | {
"line": 428,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Group α\ninst✝ : MeasurableDiv α\n⊢ Measurable Inv.inv",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Group α\ninst✝ : MeasurableDiv α\n⊢ Measurable Inv.inv"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 528,
"column": 30
} | {
"line": 528,
"column": 41
} | {
"line": 528,
"column": 42
} | [
{
"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... | [
"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 α\ninst✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 420,
"column": 29
} | {
"line": 420,
"column": 40
} | {
"line": 420,
"column": 41
} | [
{
"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",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.GDelta | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 60
} | {
"line": 81,
"column": 2
} | [
{
"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",
"ppTerm": "?in... | [
"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\ng : ℕ → ↑T\ng_surj : Surjective g\n⊢ HasSeparatingCover s... | 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": 630,
"column": 15
} | {
"line": 630,
"column": 55
} | {
"line": 630,
"column": 56
} | [
{
"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",
"ppTerm": "?m.21",
"assigned": false,
"usedConstant... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 635,
"column": 15
} | {
"line": 635,
"column": 55
} | {
"line": 635,
"column": 56
} | [
{
"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",
"ppTerm": "?m.21",
"assigned":... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1211,
"column": 4
} | {
"line": 1248,
"column": 40
} | {
"line": 1250,
"column": 0
} | [
{
"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)",
"ppTerm": "?refine_2",
"assigned": true,
"use... | [] | 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.MeasureSpace | {
"line": 1211,
"column": 4
} | {
"line": 1248,
"column": 40
} | {
"line": 1250,
"column": 0
} | [
{
"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)",
"ppTerm": "?refine_2",
"assigned": true,
"use... | [] | 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": 1291,
"column": 4
} | {
"line": 1291,
"column": 37
} | {
"line": 1291,
"column": 38
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : NeZero μ\nh : IsEmpty α\n⊢ False",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : NeZero μ\nh : IsEmpty α\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1342,
"column": 25
} | {
"line": 1342,
"column": 58
} | {
"line": 1342,
"column": 59
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
... | [
"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 ≤ ∑' (i : ι), (μ i) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 786,
"column": 2
} | {
"line": 786,
"column": 41
} | {
"line": 786,
"column": 42
} | [
{
"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",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"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 ↦ l.prod x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 791,
"column": 2
} | {
"line": 791,
"column": 41
} | {
"line": 791,
"column": 42
} | [
{
"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",
"ppTerm": "?m.22",
"assigned": true,
"usedCons... | [
"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 ↦ l.prod x) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 804,
"column": 2
} | {
"line": 804,
"column": 13
} | {
"line": 804,
"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\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))",
... | [
"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 l.prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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