module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 117,
"column": 33
} | {
"line": 117,
"column": 44
} | {
"line": 117,
"column": 45
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nf : β → Set α\ns : Set β\nhs : s.Countable\nh : ∀ b ∈ s, MeasurableSet (f b)\nthis : Countable ↑s\n⊢ ∀ (b : ↑s), MeasurableSet (f ↑b)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"Su... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nf : β → Set α\ns : Set β\nhs : s.Countable\nh : ∀ b ∈ s, MeasurableSet (f b)\nthis : Countable ↑s\n⊢ ∀ a ∈ s, MeasurableSet (f a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 34
} | {
"line": 158,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : Set (Set α)\nhs : s.Countable\nh : ∀ t ∈ s, MeasurableSet t\n⊢ MeasurableSet (⋂₀ s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Set.iInter",
"Membership.mem",
"... | [] | rw [sInter_eq_biInter]
exact MeasurableSet.biInter hs h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 34
} | {
"line": 158,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : Set (Set α)\nhs : s.Countable\nh : ∀ t ∈ s, MeasurableSet t\n⊢ MeasurableSet (⋂₀ s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Set.iInter",
"Membership.mem",
"... | [] | rw [sInter_eq_biInter]
exact MeasurableSet.biInter hs h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 12
} | {
"line": 55,
"column": 13
} | [
{
"pp": "case coe\nα : Type u_6\ninst✝ : MeasurableSpace α\nf : α → ℕ∞\nh : ∀ (n : ℕ), MeasurableSet (f ⁻¹' {↑n})\nn : ℕ\n⊢ MeasurableSet (f ⁻¹' {↑n})",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h",
"n"
],
"usedGoals": []
}
] | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 20
} | {
"line": 96,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nf : α → Prop\nh : MeasurableSet (f ⁻¹' {True})\nx : Prop\nhx : x\n⊢ MeasurableSet (f ⁻¹' {x})",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"setOf",
"Set.instSing... | [
"case pos\nα : Type u_1\nmα : MeasurableSpace α\nf : α → Prop\nh : MeasurableSet (f ⁻¹' {True})\nx : Prop\nhx : x\n⊢ MeasurableSet {a | f a}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 97,
"column": 4
} | {
"line": 98,
"column": 11
} | {
"line": 98,
"column": 12
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nf : α → Prop\nh : MeasurableSet (f ⁻¹' {True})\nx : Prop\nhx : ¬x\n⊢ MeasurableSet (f ⁻¹' {x})",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"MeasurableSet",
"eq_false",
"congrArg",
... | [
"case neg\nα : Type u_1\nmα : MeasurableSpace α\nf : α → Prop\nh : MeasurableSet (f ⁻¹' {True})\nx : Prop\nhx : ¬x\n⊢ MeasurableSet {a | ¬f a}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 35
} | {
"line": 250,
"column": 36
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\nx : α\nhx : x ∈ t\n⊢ ⟨x, hx⟩ ∈ inclusion h '' Subtype.val ⁻¹' u ↔ ⟨x, hx⟩ ∈ Subtype.val ⁻¹' u ∩ Subtype.val ⁻¹' s",
"ppTerm": "?m.114",
"assigned": true,
"us... | [
"α : Type u_1\nm : MeasurableSpace α\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\nx : α\nhx : x ∈ t\n⊢ x ∈ s ∧ x ∈ u ↔ x ∈ u ∧ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 14
} | [
{
"pp": "α : Type u_6\ninst✝¹ : SeminormedRing α\ninst✝ : NormSMulClass ℤ α\nn : ℕ\n⊢ ‖↑n‖ = ↑n * ‖1‖",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_6\ninst✝¹ : SeminormedRing α\ninst✝ : NormSMulClass ℤ α\nn : ℕ\n⊢ ‖↑n‖ = ↑n * ‖1‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 13
} | {
"line": 77,
"column": 14
} | [
{
"pp": "α : Type u_6\ninst✝² : SeminormedRing α\ninst✝¹ : NormOneClass α\ninst✝ : NormSMulClass ℤ α\na : ℕ\n⊢ ‖↑a‖ = ↑a",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_6\ninst✝² : SeminormedRing α\ninst✝¹ : NormOneClass α\ninst✝ : NormSMulClass ℤ α\na : ℕ\n⊢ ‖↑a‖ = ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 322,
"column": 18
} | {
"line": 322,
"column": 46
} | {
"line": 322,
"column": 47
} | [
{
"pp": "β : Type u_2\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable β\nx y : β\nhy : y ∉ measurableAtom x\n⊢ ∃ s, x ∈ s ∧ MeasurableSet s ∧ y ∉ s",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"β : Type u_2\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable β\nx y : β\nhy : y ∉ measurableAtom x\n⊢ ∃ s, x ∈ s ∧ MeasurableSet s ∧ y ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 25
} | {
"line": 326,
"column": 4
} | [
{
"pp": "β : Type u_2\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable β\nx : β\ns : β → Set β\nhs : ∀ y ∉ measurableAtom x, x ∈ s y ∧ MeasurableSet (s y) ∧ y ∉ s y\n⊢ measurableAtom x = ⋂ y ∈ (measurableAtom x)ᶜ, s y",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm... | [
"case h₁\nβ : Type u_2\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable β\nx : β\ns : β → Set β\nhs : ∀ y ∉ measurableAtom x, x ∈ s y ∧ MeasurableSet (s y) ∧ y ∉ s y\n⊢ measurableAtom x ⊆ ⋂ y ∈ (measurableAtom x)ᶜ, s y",
"case h₂\nβ : Type u_2\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable β\nx : β\ns : β → Set β\n... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 135,
"column": 16
} | {
"line": 136,
"column": 11
} | {
"line": 136,
"column": 12
} | [
{
"pp": "E : Type u_6\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : 0 ≤ r\na : ℝ\nha : a ≥ 0\nha' : a < 2\nr' : ℝ\nhr' : r' ≥ 0\nhr'' : r' < r\n⊢ 2 * r' ≤ diam (ball x r)",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"E : Type u_6\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : 0 ≤ r\na : ℝ\nha : a ≥ 0\nha' : a < 2\nr' : ℝ\nhr' : r' ≥ 0\nhr'' : r' < r\n⊢ diam (sphere x r') ≤ diam (ball x r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 337,
"column": 26
} | {
"line": 337,
"column": 37
} | {
"line": 337,
"column": 38
} | [
{
"pp": "𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedRing 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormOneClass 𝕜'\nc : Set 𝕜'\ns : ℝ\nhs : ∀ x ∈ cᶜ, ‖x‖ ≤ s\nx : 𝕜\nhx : x ∈ (⇑(algebraMap 𝕜 𝕜') ⁻¹' c)ᶜ\n⊢ ‖x‖ ≤ s",
"ppTerm": "?m.71",
"assigned": false,
"usedCo... | [
"𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedRing 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormOneClass 𝕜'\nc : Set 𝕜'\ns : ℝ\nhs : ∀ x ∈ cᶜ, ‖x‖ ≤ s\nx : 𝕜\nhx : x ∈ (⇑(algebraMap 𝕜 𝕜') ⁻¹' c)ᶜ\n⊢ ‖x‖ ≤ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 340,
"column": 78
} | {
"line": 342,
"column": 62
} | {
"line": 344,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedRing 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormOneClass 𝕜'\n⊢ Isometry ⇑(algebraMap 𝕜 𝕜')",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.to... | [] | by
refine Isometry.of_dist_eq fun x y => ?_
rw [dist_eq_norm, dist_eq_norm, ← map_sub, norm_algebraMap'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 35
} | {
"line": 583,
"column": 36
} | [
{
"pp": "β : Type u_2\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (a : δ) → MeasurableSpace (X a)\ng : (a : δ) → β → X a\na : δ\n⊢ MeasurableSpace.comap (g a) inferInstance ≤ MeasurableSpace.comap (fun b c ↦ g c b) pi",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Meas... | [
"β : Type u_2\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (a : δ) → MeasurableSpace (X a)\ng : (a : δ) → β → X a\na : δ\n⊢ MeasurableSpace.comap (g a) inferInstance ≤\n ⨆ i, MeasurableSpace.comap (fun b c ↦ g c b) (MeasurableSpace.comap (fun b ↦ b i) (inst✝ i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 574,
"column": 4
} | {
"line": 574,
"column": 28
} | {
"line": 575,
"column": 4
} | [
{
"pp": "𝕜✝ : Type u_1\n𝕜' : Type u_2\nE✝ : Type u_3\nF : Type u_4\nα : Type u_5\n𝕜 : Type u_6\nE : Type u_7\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Norm E\ninst✝ : Module 𝕜 E\ncore : SeminormedSpace.Core 𝕜 E\nx y : E\n⊢ ‖-x + y‖ = ‖-y + x‖",
"ppTerm": "?m.76",
"assigned": false... | [
"𝕜✝ : Type u_1\n𝕜' : Type u_2\nE✝ : Type u_3\nF : Type u_4\nα : Type u_5\n𝕜 : Type u_6\nE : Type u_7\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Norm E\ninst✝ : Module 𝕜 E\ncore : SeminormedSpace.Core 𝕜 E\nx y : E\n⊢ ‖-x + y‖ = ‖-y + x‖"
] | show ‖-x + y‖ = ‖-y + x‖ | Lean.Elab.Tactic.evalShow | Lean.Parser.Tactic.show |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 780,
"column": 6
} | {
"line": 780,
"column": 17
} | {
"line": 780,
"column": 18
} | [
{
"pp": "case pos\nδ : Type u_4\nX : δ → Type u_6\ninst✝¹ : (i : δ) → MeasurableSpace (X i)\ninst✝ : DecidableEq δ\nis : List δ\nj : δ\nhj : j ∈ j :: is\n⊢ Measurable fun v ↦ v.elim hj",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.TProd.elim_self",
"cong... | [
"case pos\nδ : Type u_4\nX : δ → Type u_6\ninst✝¹ : (i : δ) → MeasurableSpace (X i)\ninst✝ : DecidableEq δ\nis : List δ\nj : δ\nhj : j ∈ j :: is\n⊢ Measurable fun v ↦ v.1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 865,
"column": 4
} | {
"line": 865,
"column": 15
} | {
"line": 865,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np : α → Prop\nh : Measurable p\n⊢ MeasurableSet {a | p a}",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : MeasurableSpace α\np : α → Prop\nh : Measurable p\n⊢ MeasurableSet {a | p a}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 13
} | {
"line": 86,
"column": 14
} | [
{
"pp": "α : 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✝ : Fintype ι\nμ : F\ns : ι → Set α\n⊢ μ (⋃ i, s i) ≤ ∑ i, μ (s i)",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : 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✝ : Fintype ι\nμ : F\ns : ι → Set α\n⊢ μ (⋃ i, s i) ≤ ∑ i, μ (s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 31
} | {
"line": 89,
"column": 32
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns t : Set α\n⊢ μ (s ∪ t) ≤ μ s + μ t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"cond",
"Eq.mpr",
"ENNReal.instAdd",
"congrArg",
"Set.instUnion",
... | [
"α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns t : Set α\n⊢ μ (⋃ b, bif b then s else t) ≤ μ s + μ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 13
} | {
"line": 95,
"column": 14
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns t : Set α\n⊢ μ s ≤ μ (s ∩ t) + μ (s \\ t)",
"ppTerm": "?m.15",
"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\ns t : Set α\n⊢ μ s ≤ μ (s ∩ t) + μ (s \\ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 17
} | {
"line": 111,
"column": 18
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\nI : Set ι\nhI : I.Countable\ns : ι → Set α\nh : ∀ i ∈ I, μ (s i) = 0\nx✝ : Countable ↑I\n⊢ μ (⋃ i ∈ I, s i) = 0",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
"... | [
"α : Type u_1\nι : Type u_2\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\nI : Set ι\nhI : I.Countable\ns : ι → Set α\nh : ∀ i ∈ I, μ (s i) = 0\nx✝ : Countable ↑I\n⊢ μ (⋃ i ∈ I, s i) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 203,
"column": 34
} | {
"line": 203,
"column": 45
} | {
"line": 203,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ✝ : Type u_2\nm✝ : OuterMeasure α\nβ : Type ?u.16\nf : α → β\nm : OuterMeasure α\ns : ℕ → Set β\nx✝ : Pairwise (Disjoint on s)\n⊢ m (f ⁻¹' ⋃ i, s i) ≤ ∑' (i : ℕ), m (f ⁻¹' s i)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAddCom... | [
"α : Type u_1\nβ✝ : Type u_2\nm✝ : OuterMeasure α\nβ : Type ?u.16\nf : α → β\nm : OuterMeasure α\ns : ℕ → Set β\nx✝ : Pairwise (Disjoint on s)\n⊢ m (⋃ i, f ⁻¹' s i) ≤ ∑' (i : ℕ), m (f ⁻¹' s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 145,
"column": 59
} | {
"line": 145,
"column": 70
} | {
"line": 145,
"column": 71
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝² : FunLike F (Set α) ℝ≥0∞\ninst✝¹ : OuterMeasureClass F α\nι : Type u_4\nμ : F\ns : ι → Set α\nl : Filter ι\ninst✝ : l.NeBot\nS : Set α := ⋃ n, s n\nh0 : Tendsto (fun k ↦ μ (S \\ s k)) l (𝓝 0)\nM : ℝ≥0∞ := ⨆ n, μ (s n)\nA : ∀ (k : ι), μ S ≤ M + μ (S \\ s k)\n⊢ Tendsto... | [
"α : Type u_1\nF : Type u_3\ninst✝² : FunLike F (Set α) ℝ≥0∞\ninst✝¹ : OuterMeasureClass F α\nι : Type u_4\nμ : F\ns : ι → Set α\nl : Filter ι\ninst✝ : l.NeBot\nS : Set α := ⋃ n, s n\nh0 : Tendsto (fun k ↦ μ (S \\ s k)) l (𝓝 0)\nM : ℝ≥0∞ := ⨆ n, μ (s n)\nA : ∀ (k : ι), μ S ≤ M + μ (S \\ s k)\n⊢ Tendsto (fun k ↦ M ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 271,
"column": 34
} | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ✝ : Type u_2\nm✝ : OuterMeasure α\nβ : Type ?u.15\nf : α → β\nm : OuterMeasure β\ns : ℕ → Set α\nx✝ : Pairwise (Disjoint on s)\n⊢ m (f '' ⋃ i, s i) ≤ ∑' (i : ℕ), m (f '' s i)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAddCommM... | [
"α : Type u_1\nβ✝ : Type u_2\nm✝ : OuterMeasure α\nβ : Type ?u.15\nf : α → β\nm : OuterMeasure β\ns : ℕ → Set α\nx✝ : Pairwise (Disjoint on s)\n⊢ m (⋃ i, f '' s i) ≤ ∑' (i : ℕ), m (f '' s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 326,
"column": 58
} | {
"line": 326,
"column": 69
} | {
"line": 326,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nma : OuterMeasure α\nmb : OuterMeasure β\nf : α → β\nh : (map f) ma ≤ mb\ns : Set β\n⊢ ((map f) ma) s ≤ ((restrict (range f)) mb) s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"IsScalarTower.right",
... | [
"α : Type u_1\nβ : Type u_3\nma : OuterMeasure α\nmb : OuterMeasure β\nf : α → β\nh : (map f) ma ≤ mb\ns : Set β\n⊢ ma (f ⁻¹' s) ≤ mb (s ∩ range f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 780,
"column": 8
} | {
"line": 780,
"column": 83
} | {
"line": 780,
"column": 84
} | [
{
"pp": "case hbc\nG : Type u_6\nH : Type u_7\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : SeminormedAddCommGroup H\ninst✝ : NormedSpace ℝ H\ns : Set G\nf : G →+ H\nhs : s ∈ 𝓝 0\nhbounded : Bornology.IsBounded (⇑f '' s)\nδ : ℝ\nhδ : δ > 0\nhUε : ball 0 δ ⊆ s\nC ε : ℝ\nhε : 0 < ε\nhC : ∀ (a : G), ‖a‖ < δ → ‖f a... | [
"case hbc\nG : Type u_6\nH : Type u_7\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : SeminormedAddCommGroup H\ninst✝ : NormedSpace ℝ H\ns : Set G\nf : G →+ H\nhs : s ∈ 𝓝 0\nhbounded : Bornology.IsBounded (⇑f '' s)\nδ : ℝ\nhδ : δ > 0\nhUε : ball 0 δ ⊆ s\nC ε : ℝ\nhε : 0 < ε\nhC : ∀ (a : G), ‖a‖ < δ → ‖f a‖ < C\nhC₀ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 42
} | {
"line": 359,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\n⊢ (map f) ⊤ = ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
"congrArg",
... | [] | rw [map_top, hf.range_eq, restrict_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 42
} | {
"line": 359,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\n⊢ (map f) ⊤ = ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
"congrArg",
... | [] | rw [map_top, hf.range_eq, restrict_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 42
} | {
"line": 359,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\n⊢ (map f) ⊤ = ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
"congrArg",
... | [] | rw [map_top, hf.range_eq, restrict_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 165,
"column": 67
} | {
"line": 165,
"column": 78
} | {
"line": 165,
"column": 79
} | [
{
"pp": "β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Icc 0 1)\n⊢ 0 ≤ 1 - ↑x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"covariant_swap_add_of_covariant_add",
"PartialOrder.toPreo... | [
"β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Icc 0 1)\n⊢ ↑x ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 167,
"column": 67
} | {
"line": 167,
"column": 78
} | {
"line": 167,
"column": 79
} | [
{
"pp": "β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Icc 0 1)\n⊢ 1 - ↑x ≤ 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"AddGroupWithOne.toAddGroup",
"covariant_swap... | [
"β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Icc 0 1)\n⊢ 0 ≤ ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 360,
"column": 66
} | {
"line": 360,
"column": 77
} | {
"line": 360,
"column": 78
} | [
{
"pp": "β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Ioo 0 1)\n⊢ 0 < 1 - ↑x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Eq.mpr",
"sub_pos._simp_1",
"Preorder.toLT"... | [
"β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Ioo 0 1)\n⊢ ↑x < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 362,
"column": 69
} | {
"line": 362,
"column": 80
} | {
"line": 362,
"column": 81
} | [
{
"pp": "β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Ioo 0 1)\n⊢ 1 - ↑x < 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Preorder.toLT",
"AddGroupWithOne.toAddGroup",
"ins... | [
"β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Ioo 0 1)\n⊢ 0 < ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 29
} | {
"line": 48,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n\nhd : Summable d\nha : Tendsto f atTop (𝓝 a)\n⊢ dist (f 0) a ≤ tsum d",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n\nhd : Summable d\nha : Tendsto f atTop (𝓝 a)\n⊢ dist (f 0) a ≤ tsum d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 29
} | {
"line": 57,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nh : Summable fun n ↦ dist (f n) (f n.succ)\nha : Tendsto f atTop (𝓝 a)\n⊢ dist (f 0) a ≤ ∑' (n : ℕ), dist (f n) (f n.succ)",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
... | [
"α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nh : Summable fun n ↦ dist (f n) (f n.succ)\nha : Tendsto f atTop (𝓝 a)\n⊢ dist (f 0) a ≤ ∑' (n : ℕ), dist (f n) (f n.succ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 13
} | {
"line": 64,
"column": 14
} | [
{
"pp": "f : ℕ → ℝ≥0\n⊢ (¬Summable fun i ↦ ↑(f i)) ↔ Tendsto (fun n ↦ ∑ i ∈ range n, (fun i ↦ ↑(f i)) i) atTop atTop",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Finset",
"PseudoMetricSpace.toUniformSpace",
"Membership.m... | [
"f : ℕ → ℝ≥0\n⊢ (¬Summable fun i ↦ ↑(f i)) ↔ Tendsto (fun n ↦ ∑ i ∈ range n, ↑(f i)) atTop atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 13
} | {
"line": 73,
"column": 14
} | [
{
"pp": "α : Type u_4\nβ : α → Type u_3\nf : (x : α) × β x → ℝ≥0\n⊢ (Summable fun i ↦ ↑(f i)) ↔\n (∀ (x : α), Summable fun y ↦ (fun i ↦ ↑(f i)) ⟨x, y⟩) ∧ Summable fun x ↦ ∑' (y : β x), (fun i ↦ ↑(f i)) ⟨x, y⟩",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Real",
"PseudoMet... | [
"α : Type u_4\nβ : α → Type u_3\nf : (x : α) × β x → ℝ≥0\n⊢ (Summable fun i ↦ ↑(f i)) ↔ (∀ (x : α), Summable fun y ↦ ↑(f ⟨x, y⟩)) ∧ Summable fun x ↦ ∑' (y : β x), ↑(f ⟨x, y⟩)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Sign.Basic | {
"line": 40,
"column": 2
} | {
"line": 42,
"column": 14
} | {
"line": 43,
"column": 2
} | [
{
"pp": "case inl\nz : ℤ\nhz : Odd z\n⊢ 0 ^ z = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"zero_zpow",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"False",
"GroupWithZero.toDivInvMonoid",
"SignType.instCommGroupWithZero",
"congrArg",
"... | [
"case inr\ns : SignType\nz : ℤ\nhz : Odd z\nhs : s ≠ 0\n⊢ s ^ z = s"
] | · rw [zero_zpow]
rintro rfl
simp at hz | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Algebra | {
"line": 665,
"column": 46
} | {
"line": 665,
"column": 57
} | {
"line": 665,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u\ninst✝³ : TopologicalSpace A\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsSemitopologicalSemiring A\nx : A\ns : Subalgebra R A\nhs : IsClosed[inst✝³] ↑s\nhx : x ∈ s\n⊢ {x} ⊆ ↑s",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants":... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u\ninst✝³ : TopologicalSpace A\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsSemitopologicalSemiring A\nx : A\ns : Subalgebra R A\nhs : IsClosed[inst✝³] ↑s\nhx : x ∈ s\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Algebra | {
"line": 688,
"column": 23
} | {
"line": 688,
"column": 34
} | {
"line": 688,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u\ninst✝³ : TopologicalSpace A\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsSemitopologicalSemiring A\nx : A\n⊢ IsClosed[inst✝³] (range Subtype.val)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLi... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u\ninst✝³ : TopologicalSpace A\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsSemitopologicalSemiring A\nx : A\n⊢ IsClosed[inst✝³] ↑(elemental R x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Sign.Defs | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 16
} | {
"line": 291,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableLT α\na : α\nh : (if a < 0 then -1 else 0) = 1\nhn : ¬0 < a\n⊢ False",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"SignType.ctorIdx",
"False",
"Preorder.toLT",
"SignType.instOne",
... | [] | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Data.EReal.Inv | {
"line": 80,
"column": 47
} | {
"line": 80,
"column": 60
} | {
"line": 80,
"column": 60
} | [
{
"pp": "case neg_left\nx✝ y✝ : EReal\nh : (x✝ * y✝).abs = x✝.abs * y✝.abs\n⊢ (x✝ * y✝).abs = (-x✝).abs * y✝.abs",
"ppTerm": "?neg_left",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal.abs",
"EReal.abs_neg",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring... | [
"case neg_left\nx✝ y✝ : EReal\nh : (x✝ * y✝).abs = x✝.abs * y✝.abs\n⊢ (x✝ * y✝).abs = x✝.abs * y✝.abs"
] | EReal.abs_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Sign.Defs | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 16
} | {
"line": 295,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableLT α\na : α\nh : (if 0 < a then 1 else if a < 0 then -1 else 0) = -1\n⊢ a < 0",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"SignType.ctorIdx",
"False",
"Preorder.toLT",
"SignType.instOn... | [
"case pos\nα : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableLT α\na : α\nh✝¹ : ¬0 < a\nh✝ : a < 0\nh : -1 = -1\n⊢ a < 0"
] | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Data.EReal.Inv | {
"line": 140,
"column": 12
} | {
"line": 140,
"column": 23
} | {
"line": 140,
"column": 24
} | [
{
"pp": "case mp.neg\nx y : EReal\nh : ↑neg * ↑x.abs ≤ ↑neg * ↑y.abs\nhs : sign x = sign y\nhy : sign y = neg\n⊢ neg = neg ∧ neg = neg ∧ y.abs ≤ x.abs",
"ppTerm": "?mp.neg",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal.abs",
"SignType.instOne",
"congrArg",
"id"... | [
"case mp.neg\nx y : EReal\nh : ↑neg * ↑x.abs ≤ ↑neg * ↑y.abs\nhs : sign x = sign y\nhy : sign y = neg\n⊢ y.abs ≤ x.abs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 46,
"column": 54
} | {
"line": 47,
"column": 55
} | {
"line": 49,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0\nr : ℝ≥0\n⊢ HasSum (fun a ↦ ↑(f a)) ↑r ↔ HasSum f r",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"congrArg",
"Finset",
"_private.Mathlib... | [] | by
simp only [HasSum, ← ofNNReal_finsetSum, tendsto_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.EReal.Inv | {
"line": 141,
"column": 20
} | {
"line": 141,
"column": 31
} | {
"line": 141,
"column": 32
} | [
{
"pp": "case mp.pos\nx y : EReal\nh : ↑pos * ↑x.abs ≤ ↑pos * ↑y.abs\nhs : sign x = sign y\nhy : sign y = pos\n⊢ pos = pos ∧ pos = pos ∧ x.abs ≤ y.abs",
"ppTerm": "?mp.pos",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal.abs",
"SignType.instOne",
"congrArg",
"Sig... | [
"case mp.pos\nx y : EReal\nh : ↑pos * ↑x.abs ≤ ↑pos * ↑y.abs\nhs : sign x = sign y\nhy : sign y = pos\n⊢ x.abs ≤ y.abs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 20
} | {
"line": 173,
"column": 21
} | [
{
"pp": "α : Type u_4\ninst✝ : Infinite α\nc : ℝ≥0∞\nhc : c ≠ 0\nA : Tendsto (fun n ↦ ↑n * c) atTop (𝓝 (∞ * c))\nn : ℕ\ns : Finset α\nhs : #s = n\n⊢ ↑n * c ≤ ∑' (x : α), c",
"ppTerm": "?m.63",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_4\ninst✝ : Infinite α\nc : ℝ≥0∞\nhc : c ≠ 0\nA : Tendsto (fun n ↦ ↑n * c) atTop (𝓝 (∞ * c))\nn : ℕ\ns : Finset α\nhs : #s = n\n⊢ ↑n * c ≤ ∑' (x : α), c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 168,
"column": 22
} | {
"line": 168,
"column": 74
} | {
"line": 168,
"column": 75
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : TopologicalSpace α\nr : α → β → Prop\nh : Semicontinuous r\nb : β\n⊢ IsOpen[inst✝] {x | r x b}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"setOf",
"Membership.mem",
"nhds",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : TopologicalSpace α\nr : α → β → Prop\nh : Semicontinuous r\nb : β\n⊢ ∀ (x : α), r x b → {x | r x b} ∈ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 18
} | {
"line": 174,
"column": 19
} | [
{
"pp": "α : Type u_4\ninst✝ : Infinite α\nc : ℝ≥0∞\nhc : c ≠ 0\nA : Tendsto (fun n ↦ ↑n * c) atTop (𝓝 (∞ * c))\nB : ∀ (n : ℕ), ↑n * c ≤ ∑' (x : α), c\n⊢ ∑' (x : α), c = ∞",
"ppTerm": "?m.52",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_4\ninst✝ : Infinite α\nc : ℝ≥0∞\nhc : c ≠ 0\nA : Tendsto (fun n ↦ ↑n * c) atTop (𝓝 (∞ * c))\nB : ∀ (n : ℕ), ↑n * c ≤ ∑' (x : α), c\n⊢ ∑' (x : α), c = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 33
} | {
"line": 193,
"column": 34
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0∞\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\na : R\n⊢ ∑' (i : α), a • f i = a • ∑' (i : α), f i",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nf : α → ℝ≥0∞\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\na : R\n⊢ ∑' (i : α), a • f i = a • ∑' (i : α), f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 85
} | {
"line": 217,
"column": 86
} | [
{
"pp": "α : Type u_4\nf : α → ℝ≥0∞\nhf : ∑' (i : α), f i ≠ ∞\n⊢ Summable (ENNReal.toNNReal ∘ f)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"congrArg",
"_priva... | [
"α : Type u_4\nf : α → ℝ≥0∞\nhf : ∑' (i : α), f i ≠ ∞\n⊢ ∑' (b : α), f b ≠ ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 287,
"column": 14
} | {
"line": 287,
"column": 25
} | {
"line": 287,
"column": 26
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0∞\ns t : Set α\n⊢ ∑' (x : ↑(⋃ b, bif b then s else t)), f ↑x ≤ ∑' (x : ↑s), f ↑x + ∑' (x : ↑t), f ↑x",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nf : α → ℝ≥0∞\ns t : Set α\n⊢ ∑' (x : ↑(⋃ b, bif b then s else t)), f ↑x ≤ ∑' (x : ↑s), f ↑x + ∑' (x : ↑t), f ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 1092,
"column": 4
} | {
"line": 1092,
"column": 15
} | {
"line": 1092,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nh : HasOpenCGraph f\nx : α\nb : β\nhb : b ∈ f x\n⊢ IsOpen[inst✝¹] {x' | b ∈ f x'}",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nh : HasOpenCGraph f\nx : α\nb : β\nhb : b ∈ f x\n⊢ IsOpen[inst✝¹] {x' | b ∈ f x'}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 1093,
"column": 2
} | {
"line": 1093,
"column": 33
} | {
"line": 1093,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nh : HasOpenCGraph f\nx : α\nb : β\nhb : b ∈ f x\nhopen : IsOpen[inst✝¹] {x' | b ∈ f x'}\n⊢ ∀ᶠ (x' : α) in 𝓝 x, (fun x b ↦ b ∈ f x) x' b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants"... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nh : HasOpenCGraph f\nx : α\nb : β\nhb : b ∈ f x\nhopen : IsOpen[inst✝¹] {x' | b ∈ f x'}\n⊢ {x | b ∈ f x} ∈ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 23
} | {
"line": 380,
"column": 24
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0\nh : Summable fun a ↦ ↑(f a)\n⊢ (support f).Countable",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"id",
"NNReal",
"NNReal.instZero",
"Function.support",
"Set.Countable"
],
"usedFVars": [
"α",
"f"
]... | [
"α : Type u_1\nf : α → ℝ≥0\nh : Summable fun a ↦ ↑(f a)\n⊢ {x | ¬f x = 0}.Countable"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 423,
"column": 4
} | {
"line": 424,
"column": 36
} | {
"line": 424,
"column": 37
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : α → Type u_4\nf : (x : α) × β x → ℝ≥0\nh₁ : ∀ (x : α), Summable fun y ↦ f ⟨x, y⟩\nh₂ : Summable fun x ↦ ∑' (y : β x), f ⟨x, y⟩\n⊢ Summable f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNRe... | [
"case mpr\nα : Type u_1\nβ : α → Type u_4\nf : (x : α) × β x → ℝ≥0\nh₁ : ∀ (x : α), Summable fun y ↦ f ⟨x, y⟩\nh₂ : Summable fun x ↦ ∑' (y : β x), f ⟨x, y⟩\n⊢ ∑' (a : α) (b : β a), ↑(f ⟨a, b⟩) ≠ ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 13
} | {
"line": 471,
"column": 14
} | [
{
"pp": "α : Type u_1\ng : α → ℝ≥0\nhg : Summable g\ni : α\nhi : 0 < g i\n⊢ 0 < ∑' (b : α), g b",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ng : α → ℝ≥0\nhg : Summable g\ni : α\nhi : 0 < g i\n⊢ 0 < ∑' (b : α), g b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Inv | {
"line": 395,
"column": 42
} | {
"line": 395,
"column": 77
} | {
"line": 395,
"column": 77
} | [
{
"pp": "case refine_2\na b c : EReal\nhbot : b ≠ ⊥\nhtop : b ≠ ⊤\nhzero : b ≠ 0\nh : c = a * b\n⊢ b * (a / b) = a",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal.instDivInvMonoid",
"instHDiv",
"HMul.hMul",
"congrArg",
"EReal",
... | [
"case refine_2\na b c : EReal\nhbot : b ≠ ⊥\nhtop : b ≠ ⊤\nhzero : b ≠ 0\nh : c = a * b\n⊢ a = a"
] | @mul_div_cancel a b hbot htop hzero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 23
} | {
"line": 554,
"column": 24
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0\nh : ∑' (i : α), (fun i ↦ ↑(f i)) i ≠ ∞\n⊢ (support fun i ↦ ↑(f i)).Countable",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"congrArg",
"setOf",
"id",
"NNReal",
"Ne",
"NNReal.... | [
"α : Type u_1\nf : α → ℝ≥0\nh : ∑' (i : α), (fun i ↦ ↑(f i)) i ≠ ∞\n⊢ {x | ¬f x = 0}.Countable"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Inv | {
"line": 447,
"column": 2
} | {
"line": 449,
"column": 63
} | {
"line": 451,
"column": 0
} | [
{
"pp": "a b c : EReal\nh : 0 < b\nh' : b ≠ ⊤\n⊢ a / b ≤ c ↔ a ≤ b * c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"EReal.instDivInvMonoid",
"instHDiv",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"ER... | [] | nth_rw 1 [← @mul_div_cancel c b (ne_bot_of_gt h) h' h.ne']
rw [mul_div b c b, mul_comm b]
exact StrictMono.le_iff_le (strictMono_div_right_of_pos h h') | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Inv | {
"line": 447,
"column": 2
} | {
"line": 449,
"column": 63
} | {
"line": 451,
"column": 0
} | [
{
"pp": "a b c : EReal\nh : 0 < b\nh' : b ≠ ⊤\n⊢ a / b ≤ c ↔ a ≤ b * c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"EReal.instDivInvMonoid",
"instHDiv",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"ER... | [] | nth_rw 1 [← @mul_div_cancel c b (ne_bot_of_gt h) h' h.ne']
rw [mul_div b c b, mul_comm b]
exact StrictMono.le_iff_le (strictMono_div_right_of_pos h h') | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 153,
"column": 70
} | {
"line": 165,
"column": 43
} | {
"line": 167,
"column": 0
} | [
{
"pp": "⊢ 𝓝[≠] ⊥ = map Real.toEReal atBot",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Set.Ioc",
"False",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"Real.instArch... | [] | by
apply (nhdsWithin_hasBasis nhds_bot_basis_Iic _).ext (atBot_basis.map Real.toEReal)
· simp only [EReal.image_coe_Iic,
true_and]
intro x hx
by_cases hx_top : x = ⊤
· simp [hx_top]
lift x to ℝ using ⟨hx_top, hx.ne_bot⟩
refine ⟨x, fun x ⟨h1, h2⟩ ↦ ?_⟩
simp [h2, h1.ne_bot]
· simp only... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 83
} | {
"line": 209,
"column": 6
} | [
{
"pp": "case pos\nx✝ : EReal\nh_top : ¬x✝ = ⊤\nx : EReal\nhx : x ∈ {⊤}ᶜ\nh_bot : x = ⊥\n⊢ ∃ i, True ∧ ∀ ⦃x : EReal⦄, x ∈ Iio ↑i → ENNReal.ofReal x.toReal = (fun x ↦ ENNReal.ofReal x.toReal) ⊥",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real"... | [
"case pos\nx✝ : EReal\nh_top : ¬x✝ = ⊤\nx : EReal\nhx : x ∈ {⊤}ᶜ\nh_bot : x = ⊥\n⊢ ∃ i, ∀ ⦃x : EReal⦄, x < ↑i → x.toReal ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 627,
"column": 36
} | {
"line": 627,
"column": 47
} | {
"line": 627,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\n⊢ edist (f 0) a ≤ ∑' (m : ℕ), d m",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
... | [
"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\n⊢ edist (f 0) a ≤ ∑' (m : ℕ), d m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 646,
"column": 20
} | {
"line": 646,
"column": 45
} | {
"line": 646,
"column": 46
} | [
{
"pp": "α : Type u_4\n⊢ ∑' (x : ↑univ), 1 = ↑(ENat.card α)",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_4\n⊢ ∑' (x : ↑univ), 1 = ↑(ENat.card α)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 650,
"column": 20
} | {
"line": 650,
"column": 45
} | {
"line": 650,
"column": 46
} | [
{
"pp": "α : Type u_4\nc : ℝ≥0∞\n⊢ ∑' (x : ↑univ), c = ↑(ENat.card α) * c",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_4\nc : ℝ≥0∞\n⊢ ∑' (x : ↑univ), c = ↑(ENat.card α) * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 11
} | {
"line": 311,
"column": 12
} | [
{
"pp": "case inr.a\nα : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ ∀ y > limsup u f * c, ∀ᶠ (a : α) in f, u a * c < y",
"ppTerm": "?inr.a",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal.instDivInvMonoid",
"False... | [
"case inr.a\nα : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ ∀ (y : EReal), limsup u f < y / c → ∀ᶠ (a : α) in f, u a < y / c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 11
} | {
"line": 314,
"column": 12
} | [
{
"pp": "case a\nα : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ ∀ y < limsup u f * c, ∃ᶠ (a : α) in f, y < u a * c",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal.instDivInvMonoid",
"False",
... | [
"case a\nα : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ ∀ (y : EReal), y / c < limsup u f → ∃ᶠ (a : α) in f, y / c < u a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 26
} | {
"line": 319,
"column": 4
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ limsup (fun x ↦ c * u x) f = c * liminf u f",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ limsup (fun x ↦ c * u x) f = c * liminf u f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 49
} | {
"line": 324,
"column": 4
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ liminf (fun x ↦ c * u x) f = c * liminf u f",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ liminf (fun x ↦ c * u x) f = c * liminf u f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 49
} | {
"line": 329,
"column": 4
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ liminf (fun x ↦ c * u x) f = c * limsup u f",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ liminf (fun x ↦ c * u x) f = c * limsup u f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 75,
"column": 20
} | {
"line": 75,
"column": 31
} | {
"line": 75,
"column": 32
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nμ : Set α → ℝ≥0∞ := fun s ↦ ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)\ns : ℕ → Set α\nx✝ : Pairwise (Disjoint on s)\nε : ℝ≥0\nhε : 0 < ε\nhb : ∑' (i : ℕ), μ (s i) < ∞\nε' : ℕ → ℝ≥0\nhε' : ∀ (i : ℕ), 0 < ε' i\nhl : ∑' (i : ℕ), ↑(ε' i) < ↑ε\ni : ℕ\n... | [
"α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nμ : Set α → ℝ≥0∞ := fun s ↦ ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)\ns : ℕ → Set α\nx✝ : Pairwise (Disjoint on s)\nε : ℝ≥0\nhε : 0 < ε\nhb : ∑' (i : ℕ), μ (s i) < ∞\nε' : ℕ → ℝ≥0\nhε' : ∀ (i : ℕ), 0 < ε' i\nhl : ∑' (i : ℕ), ↑(ε' i) < ↑ε\ni : ℕ\n⊢ ¬ε' i = 0"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 45
} | {
"line": 54,
"column": 4
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : DivisionSemiring 𝕜\ninst✝³ : CharZero 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : ContinuousSMul ℚ≥0 𝕜\ninst✝ : ContinuousMul 𝕜\nC : 𝕜\n⊢ Tendsto (fun n ↦ C / ↑n) atTop (𝓝 0)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAsso... | [
"𝕜 : Type u_4\ninst✝⁴ : DivisionSemiring 𝕜\ninst✝³ : CharZero 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : ContinuousSMul ℚ≥0 𝕜\ninst✝ : ContinuousMul 𝕜\nC : 𝕜\n⊢ Tendsto (fun n ↦ C * (↑n)⁻¹) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 50
} | {
"line": 325,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ns : Set α\nγ : Type u_4\ninst✝ : LinearOrder γ\nι : Type u_5\nf : ι → α → γ\nks : IsCompact s\nI : Set ι\nc : γ\nhfi : ∀ i ∈ I, LowerSemicontinuousOn (f i) s\nH : s ∩ ⋂ i ∈ I, f i ⁻¹' Iic c = ∅\nthis : ∀ i ∈ I, IsClosed[instTopologicalSpaceSubtype] (s ↓∩ (fun ... | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\ns : Set α\nγ : Type u_4\ninst✝ : LinearOrder γ\nι : Type u_5\nf : ι → α → γ\nks : IsCompact s\nI : Set ι\nc : γ\nhfi : ∀ i ∈ I, LowerSemicontinuousOn (f i) s\nH : s ∩ ⋂ i ∈ I, f i ⁻¹' Iic c = ∅\nthis : ∀ i ∈ I, IsClosed[instTopologicalSpaceSubtype] (s ↓∩ (fun i ↦ f i ⁻¹' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 58
} | {
"line": 360,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_3\nf : Filter α\nu v : α → EReal\nhu : ∃ᶠ (x : α) in f, 0 ≤ u x\nhv : 0 ≤ᶠ[f] v\nh✝ : f.NeBot\nu_0 : 0 ≤ limsup u f\nh₁ : 0 < limsup u f ∨ limsup v f ≠ ⊤\nh₂ : limsup u f ≠ ⊤ ∨ 0 < limsup v f\na : EReal\na_u : a > limsup u f\nb : EReal\nb_v : b > limsup v f\nc : EReal\nc_ab : c > a... | [] | exact mul_nonneg (u_0.trans a_u.le) (v_0.trans x_b.le) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 35
} | {
"line": 108,
"column": 4
} | [
{
"pp": "case h\n𝕜 : Type u_4\ninst✝⁵ : Semifield 𝕜\ninst✝⁴ : CharZero 𝕜\ninst✝³ : TopologicalSpace 𝕜\ninst✝² : ContinuousSMul ℚ≥0 𝕜\ninst✝¹ : IsTopologicalSemiring 𝕜\ninst✝ : ContinuousInv₀ 𝕜\na b c d : 𝕜\nhd : d ≠ 0\n⊢ Tendsto (fun k ↦ (a * (↑k)⁻¹ + c) / (b * (↑k)⁻¹ + d)) atTop (𝓝 (c / d))",
"ppT... | [
"𝕜 : Type u_4\ninst✝⁵ : Semifield 𝕜\ninst✝⁴ : CharZero 𝕜\ninst✝³ : TopologicalSpace 𝕜\ninst✝² : ContinuousSMul ℚ≥0 𝕜\ninst✝¹ : IsTopologicalSemiring 𝕜\ninst✝ : ContinuousInv₀ 𝕜\na b c d : 𝕜\nhd : d ≠ 0\n⊢ Tendsto (fun k ↦ a * (↑k)⁻¹ + c) atTop (𝓝 c)",
"𝕜 : Type u_4\ninst✝⁵ : Semifield 𝕜\ninst✝⁴ : CharZ... | apply Filter.Tendsto.div _ _ hd | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 53
} | {
"line": 399,
"column": 54
} | [
{
"pp": "a r : ℝ\nx✝ : EReal × EReal\nh : ↑(r - (a - 1)) < x✝.1 ∧ ↑(a - 1) < x✝.2\n⊢ ↑r < x✝.1 + x✝.2",
"ppTerm": "?m.67",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a r : ℝ\nx✝ : EReal × EReal\nh : ↑(r - (a - 1)) < x✝.1 ∧ ↑(a - 1) < x✝.2\n⊢ ↑r < x✝.1 + x✝.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 403,
"column": 2
} | {
"line": 404,
"column": 9
} | {
"line": 404,
"column": 10
} | [
{
"pp": "a : ℝ\n⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a, ⊤)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"ContinuousAt",
"EReal.instTopologicalSpace",
"instTopologicalSpaceProd",
"EReal",
"instTopEReal",
"id",
"Prod.mk",
"instAddCommMonoidE... | [
"a : ℝ\n⊢ Tendsto (fun p ↦ p.1 + p.2) (𝓝 (↑a, ⊤)) (𝓝 (↑a + ⊤))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 39
} | {
"line": 410,
"column": 40
} | [
{
"pp": "r : ℝ\nx✝ : EReal × EReal\nh : ↑0 < x✝.1 ∧ ↑r < x✝.2\n⊢ ↑r < x✝.1 + x✝.2",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\nx✝ : EReal × EReal\nh : ↑0 < x✝.1 ∧ ↑r < x✝.2\n⊢ ↑r < x✝.1 + x✝.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 53
} | {
"line": 417,
"column": 54
} | [
{
"pp": "a r : ℝ\nx✝ : EReal × EReal\nh : x✝.1 < ↑(r - (a + 1)) ∧ x✝.2 < ↑(a + 1)\n⊢ x✝.1 + x✝.2 < ↑r",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a r : ℝ\nx✝ : EReal × EReal\nh : x✝.1 < ↑(r - (a + 1)) ∧ x✝.2 < ↑(a + 1)\n⊢ x✝.1 + x✝.2 < ↑r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 421,
"column": 2
} | {
"line": 422,
"column": 9
} | {
"line": 422,
"column": 10
} | [
{
"pp": "a : ℝ\n⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a, ⊥)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"ContinuousAt",
"EReal.instTopologicalSpace",
"instTopologicalSpaceProd",
"EReal",
"nhds",
"id",
"Prod.mk",... | [
"a : ℝ\n⊢ Tendsto (fun p ↦ p.1 + p.2) (𝓝 (↑a, ⊥)) (𝓝 (⊥ + ↑a))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 39
} | {
"line": 428,
"column": 40
} | [
{
"pp": "r : ℝ\nx✝ : EReal × EReal\nh : x✝.1 < ↑0 ∧ x✝.2 < ↑r\n⊢ x✝.1 + x✝.2 < ↑r",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\nx✝ : EReal × EReal\nh : x✝.1 < ↑0 ∧ x✝.2 < ↑r\n⊢ x✝.1 + x✝.2 < ↑r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 451,
"column": 10
} | {
"line": 451,
"column": 25
} | {
"line": 451,
"column": 26
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | [
"case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 455,
"column": 10
} | {
"line": 455,
"column": 25
} | {
"line": 455,
"column": 26
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | [
"case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 465,
"column": 10
} | {
"line": 465,
"column": 25
} | {
"line": 465,
"column": 26
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | [
"case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 232,
"column": 2
} | {
"line": 236,
"column": 44
} | {
"line": 238,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ c • OuterMeasure.ofFunction m m_empty = OuterMeasure.ofFunction (c • m) ⋯",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MeasureTheory.OuterMeasure.instIsSMulApplySetENNReal",
"Eq.mpr",
... | [] | ext1 s
haveI : Nonempty { t : ℕ → Set α // s ⊆ ⋃ i, t i } := ⟨⟨fun _ => s, subset_iUnion (fun _ => s) 0⟩⟩
simp only [smul_apply, ofFunction_apply, ENNReal.tsum_mul_left, Pi.smul_apply, smul_eq_mul,
iInf_subtype']
rw [ENNReal.mul_iInf fun h => (hc h).elim] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 232,
"column": 2
} | {
"line": 236,
"column": 44
} | {
"line": 238,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ c • OuterMeasure.ofFunction m m_empty = OuterMeasure.ofFunction (c • m) ⋯",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MeasureTheory.OuterMeasure.instIsSMulApplySetENNReal",
"Eq.mpr",
... | [] | ext1 s
haveI : Nonempty { t : ℕ → Set α // s ⊆ ⋃ i, t i } := ⟨⟨fun _ => s, subset_iUnion (fun _ => s) 0⟩⟩
simp only [smul_apply, ofFunction_apply, ENNReal.tsum_mul_left, Pi.smul_apply, smul_eq_mul,
iInf_subtype']
rw [ENNReal.mul_iInf fun h => (hc h).elim] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 269,
"column": 15
} | {
"line": 269,
"column": 78
} | {
"line": 270,
"column": 4
} | [
{
"pp": "r : ℝ≥0\nh : Tendsto (fun n ↦ r ^ n) atTop (𝓝 0)\n⊢ r < 1",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ≥0\nh : Tendsto (fun n ↦ r ^ n) atTop (𝓝 0)\n⊢ r < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 35
} | {
"line": 303,
"column": 36
} | [
{
"pp": "case refine_2\nr : ℝ≥0∞\nr_gt_one : 1 < r\nobs : r⁻¹ < 1 → Tendsto (fun x ↦ (r⁻¹ ^ x)⁻¹) atTop (𝓝 ∞)\n⊢ Tendsto (fun n ↦ r ^ n) atTop (𝓝 ∞)",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nr : ℝ≥0∞\nr_gt_one : 1 < r\nobs : r⁻¹ < 1 → Tendsto (fun x ↦ (r⁻¹ ^ x)⁻¹) atTop (𝓝 ∞)\n⊢ Tendsto (fun n ↦ r ^ n) atTop (𝓝 ∞)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 478,
"column": 10
} | {
"line": 478,
"column": 25
} | {
"line": 478,
"column": 26
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | [
"case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 479,
"column": 10
} | {
"line": 479,
"column": 28
} | {
"line": 479,
"column": 29
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | [
"case neg\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 45
} | {
"line": 475,
"column": 46
} | [
{
"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)",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 70
} | {
"line": 505,
"column": 4
} | [
{
"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",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 15
} | {
"line": 522,
"column": 16
} | [
{
"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",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 431,
"column": 4
} | {
"line": 432,
"column": 29
} | {
"line": 432,
"column": 30
} | [
{
"pp": "case refine_2\nα : Type u_1\nι : Sort u_2\nβ : Type u_3\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\nn : ℕ\ni : ι\n⊢ f '' f ⁻¹' (fun n ↦ f '' t n ∪ (range f)ᶜ) n ⊆ f '' t n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstant... | [
"case refine_2\nα : Type u_1\nι : Sort u_2\nβ : Type u_3\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\nn : ℕ\ni : ι\n⊢ f ⁻¹' f '' t n ⊆ f ⁻¹' f '' t n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 554,
"column": 2
} | {
"line": 561,
"column": 61
} | {
"line": 563,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na : Finset ι\nha : ∀ i ∈ a, LowerSemic... | [] | classical
induction a using Finset.induction_on with
| empty => exact lowerSemicontinuousWithinAt_const
| insert _ _ ia IH =>
simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self ..))
(IH fun j ja => ha j (Finset.m... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 554,
"column": 2
} | {
"line": 561,
"column": 61
} | {
"line": 563,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na : Finset ι\nha : ∀ i ∈ a, LowerSemic... | [] | classical
induction a using Finset.induction_on with
| empty => exact lowerSemicontinuousWithinAt_const
| insert _ _ ia IH =>
simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self ..))
(IH fun j ja => ha j (Finset.m... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 554,
"column": 2
} | {
"line": 561,
"column": 61
} | {
"line": 563,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na : Finset ι\nha : ∀ i ∈ a, LowerSemic... | [] | classical
induction a using Finset.induction_on with
| empty => exact lowerSemicontinuousWithinAt_const
| insert _ _ ia IH =>
simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self ..))
(IH fun j ja => ha j (Finset.m... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SetDissipate | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 23
} | {
"line": 57,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\ns : α → Set β\nx : α\n⊢ ⋂ y, ⋂ (_ : y ≤ x), dissipate s y = dissipate s x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Set.dissipate",
"Set.iInter",
"Preorder.toLE",
"LE.le"
... | [
"case h₁\nα : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\ns : α → Set β\nx : α\n⊢ ⋂ y, ⋂ (_ : y ≤ x), dissipate s y ⊆ dissipate s x",
"case h₂\nα : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\ns : α → Set β\nx : α\n⊢ dissipate s x ⊆ ⋂ y, ⋂ (_ : y ≤ x), dissipate s y"
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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