module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 20
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), f a ≤ 1\nh'f : ∀ a ∈ sᶜ, f a = 0\nx : α\nhx : x ∈ s\n⊢ f x ≤ s.indicator 1 x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.indicator",
"id",
"LE.le",
"Set.in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 20
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), f a ≤ 1\nh'f : ∀ a ∈ sᶜ, f a = 0\nx : α\nhx : x ∉ s\n⊢ f x ≤ s.indicator 1 x",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"False",
"eq_false",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 337,
"column": 26
} | {
"line": 337,
"column": 37
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 340,
"column": 78
} | {
"line": 342,
"column": 62
} | [
{
"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')",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminorm... | 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.Analysis.Normed.Module.Basic | {
"line": 574,
"column": 4
} | {
"line": 574,
"column": 28
} | [
{
"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‖",
"usedConstants": []
}
] | show ‖-x + y‖ = ‖-y + x‖ | Lean.Elab.Tactic.evalShow | Lean.Parser.Tactic.show |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 38
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᶠ[ae μ] g\nhf : ∫⁻ (x : α), f x ∂μ ≠ ∞\nhg : AEMeasurable g μ\nhgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 13
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhμ : ¬μ univ = 0\nhg : AEMeasurable g μ\nhfi : ∫⁻ (x : α), f x ∂μ ≠ ∞\nh : ∀ᵐ (x : α) ∂μ, f x < g x\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ univ → f x < g x",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 780,
"column": 8
} | {
"line": 780,
"column": 83
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 72,
"column": 14
} | {
"line": 74,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nh_meas : ∀ (n : ℕ), Measurable (f n)\nh_mono✝ : ∀ (n : ℕ), f n.succ ≤ᶠ[ae μ] f n\nh_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞\nfn_le_f0 : ∫⁻ (a : α), ⨅ n, f n a ∂μ ≤ ∫⁻ (a : α), f 0 a ∂μ\nfn_le_f0' : ⨅ n, ∫⁻ (a : α), f n a ∂μ ≤ ∫⁻ (a : α), ... | induction n with
| zero => rfl
| succ n ih => exact (h n).trans ih | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 72,
"column": 14
} | {
"line": 74,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nh_meas : ∀ (n : ℕ), Measurable (f n)\nh_mono✝ : ∀ (n : ℕ), f n.succ ≤ᶠ[ae μ] f n\nh_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞\nfn_le_f0 : ∫⁻ (a : α), ⨅ n, f n a ∂μ ≤ ∫⁻ (a : α), f 0 a ∂μ\nfn_le_f0' : ⨅ n, ∫⁻ (a : α), f n a ∂μ ≤ ∫⁻ (a : α), ... | induction n with
| zero => rfl
| succ n ih => exact (h n).trans ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 72,
"column": 14
} | {
"line": 74,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nh_meas : ∀ (n : ℕ), Measurable (f n)\nh_mono✝ : ∀ (n : ℕ), f n.succ ≤ᶠ[ae μ] f n\nh_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞\nfn_le_f0 : ∫⁻ (a : α), ⨅ n, f n a ∂μ ≤ ∫⁻ (a : α), f 0 a ∂μ\nfn_le_f0' : ⨅ n, ∫⁻ (a : α), f n a ∂μ ≤ ∫⁻ (a : α), ... | induction n with
| zero => rfl
| succ n ih => exact (h n).trans ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 55
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : ∫⁻ (a : α), f a ∂μ ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nhf₀ : ¬∫⁻ (a : α), f a ∂μ = 0\n⊢ ∃ g, ∃ (_ : ⇑g ≤ f), ∫⁻ (a : α), f a ∂μ - ε < g.lintegral μ",
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.Lebesgue.Sub.0.Me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 41
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl : List (α → M)\nhl : ∀ f ∈ l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x ↦ (List.map (fun f ↦ f x) l).prod) μ",
"usedConstants": [
"Eq.mpr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 13
} | [
{
"pp": "case mk\nα : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (Multiset.p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 433,
"column": 46
} | {
"line": 433,
"column": 57
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, AEStronglyMeasurable f μ\n⊢ ∀ f ∈ l, AEStronglyMeasurable f μ",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 439,
"column": 2
} | {
"line": 439,
"column": 45
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\ns : Multiset (α → M)\nhs : ∀ f ∈ s, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x ↦ (Multiset.map (fun f ↦ f x) s).prod) μ",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 40
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nι : Type u_6\nf : ι → α → M\ns : Finset ι\nhf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ\n⊢ AEStronglyMeasurable (fun a ↦ ∏ i ∈ s, f i a) μ",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 620,
"column": 2
} | {
"line": 620,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nmβ : MeasurableSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nf : α → β\nhf : AEStronglyMeasurable f μ\nt : Set β\nht1 : IsSeparable t\nht2 : ∀ᵐ (x : α) ∂μ, f x ∈ t\n⊢ (Measure.map f μ) (cl... | refine ae_map_iff hf.aemeasurable isClosed_closure.measurableSet |>.2 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 189,
"column": 6
} | {
"line": 191,
"column": 22
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\nh : Disjoint μ ν\nε : ℝ≥0\nhε : 0 < ε\nh₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0\nn : ℕ\n⊢ ∃ x ∈ {m | ∃ t, m = μ t + ν tᶜ}, x < ↑ε * (1 / 2) ^ n",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.... | refine exists_lt_of_csInf_lt ⟨ν univ, ∅, by simp⟩ <| h₁ ▸ ENNReal.mul_pos ?_ (by simp)
norm_cast
exact hε.ne.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 189,
"column": 6
} | {
"line": 191,
"column": 22
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\nh : Disjoint μ ν\nε : ℝ≥0\nhε : 0 < ε\nh₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0\nn : ℕ\n⊢ ∃ x ∈ {m | ∃ t, m = μ t + ν tᶜ}, x < ↑ε * (1 / 2) ^ n",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.... | refine exists_lt_of_csInf_lt ⟨ν univ, ∅, by simp⟩ <| h₁ ▸ ENNReal.mul_pos ?_ (by simp)
norm_cast
exact hε.ne.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 31
} | [
{
"pp": "case hbc\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nhf : StronglyMeasurable f\nc : ℝ\nx : α\nhfx : ‖f x‖ ≤ c\nh_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)\nhfx0 : f x = 0\nh_tendsto_norm : Tendsto (fun n ↦ ‖(hf.ap... | · exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 183,
"column": 14
} | {
"line": 183,
"column": 30
} | [
{
"pp": "case h.zero\nα : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhF_meas : AEMeasurable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 722,
"column": 4
} | {
"line": 722,
"column": 37
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\nmb : MeasurableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nμ : Measure α\nf : α → β\ns : Set β\nhs : MeasurableSet s\nh_nonempty : s.Nonempty\nh_mem : ∀ᵐ (x : α) ∂μ, f x ∈ s\nf' : α → β\nhf' : StronglyM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 726,
"column": 61
} | {
"line": 726,
"column": 78
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\nmb : MeasurableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nμ : Measure α\nf : α → β\ns : Set β\nhs : MeasurableSet s\nh_nonempty : s.Nonempty\nh_mem : ∀ᵐ (x : α) ∂μ, f x ∈ s\nf' : α → β\nhf' : StronglyM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 42
} | [
{
"pp": "case neg.inl\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nc : ℝ\nhf : StronglyMeasurable f\nhc : 0 ≤ c\nn : ℕ\nx : α\nh0 : ¬‖(hf.approx n) x‖ = 0\nh : ‖(hf.approx n) x‖ ≤ c\n⊢ ‖1‖ * ‖(hf.approx n) x‖ ≤ c",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 823,
"column": 15
} | {
"line": 823,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nf : α → β\nG : Type u_6\ninst✝² : Group G\ninst✝¹ : MulAction G β\ninst✝ : ContinuousConstSMul G β\nc : G\nh : AEStronglyMeasurable (fun x ↦ c • f x) μ\n⊢ AEStronglyMeasurable f μ",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 48
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : MeasurableSet s\n⊢ μ.real s + μ.real sᶜ = 1",
"usedConstants": [
"Real",
"Compl.compl",
"MeasureTheory.Measure.real",
"id",
"Set.instCompl",
"Real.instAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 51
} | [
{
"pp": "case h\nα : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ... | exact ge_of_tendsto' h_tendsto (fun m ↦ h_le _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 52
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : α → Prop\nf✝ : β → α\nf : α → β\nhf : AEMeasurable f μ\n⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 46
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : ℝ≥0∞\nhμs : p ≤ μ s\ns_mble : MeasurableSet s\nh : IsProbabilityMeasure μ\n⊢ μ sᶜ ≤ 1 - p",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"MeasureTheory.Measure",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UnitInterval | {
"line": 173,
"column": 57
} | {
"line": 173,
"column": 68
} | [
{
"pp": "x : ↑I\n⊢ 0 ≤ 1 - ↑x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"HSub.hSub",
"Membership.mem",
"id",
"Real.instAddGroup",
"LE.le",
"Real.instAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UnitInterval | {
"line": 178,
"column": 57
} | {
"line": 178,
"column": 68
} | [
{
"pp": "x : ↑I\n⊢ 1 - ↑x ≤ 1",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Real.instAddMonoid",
"instIsLeftCancelAddOfAddLeftReflectLE",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"AddMonoid.toAddZeroClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UnitInterval | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 13
} | [
{
"pp": "i j : ↑I\nh : 0 < ↑i ∧ ↑j < 1\n⊢ ↑j * ↑i < ↑i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 534,
"column": 15
} | {
"line": 534,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nG : Type u_6\ninst✝³ : TopologicalSpace β\ninst✝² : Group G\ninst✝¹ : MulAction G β\ninst✝ : ContinuousConstSMul G β\nm : MeasurableSpace α\nc : G\nh : StronglyMeasurable fun x ↦ c • f x\n⊢ StronglyMeasurable f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 57,
"column": 48
} | {
"line": 57,
"column": 86
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : Set α\ns_fin : s.Finite\ns_mble : MeasurableSet s\n⊢ MeasurableSet ↑s_fin.toFinset",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Finset",
"id",
"Set.Finite.coe_toFinset",
"SetLike.coe",
"Fins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 41
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl : ∀ f ∈ l, StronglyMeasurable f\n⊢ StronglyMeasurable fun x ↦ (List.map (fun f ↦ f x) l).prod",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 605,
"column": 2
} | {
"line": 605,
"column": 13
} | [
{
"pp": "case mk\nα : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, StronglyMeasurable f\n⊢ StronglyMeasurable (Multiset.prod (Quot.mk (⇑(List.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 605,
"column": 44
} | {
"line": 605,
"column": 55
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : Multiset (α → M)\nl : List (α → M)\nhl : ∀ f ∈ Quot.mk (⇑(List.isSetoid (α → M))) l, StronglyMeasurable f\n⊢ ∀ f ∈ l, StronglyMeasurable f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 45
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\ns : Multiset (α → M)\nhs : ∀ f ∈ s, StronglyMeasurable f\n⊢ StronglyMeasurable fun x ↦ (Multiset.map (fun f ↦ f x) s).prod",
"usedConstants": [
"Eq.mpr",
"Mult... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 621,
"column": 2
} | {
"line": 621,
"column": 40
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nι : Type u_6\nf : ι → α → M\ns : Finset ι\nhf : ∀ i ∈ s, StronglyMeasurable (f i)\n⊢ StronglyMeasurable fun a ↦ ∏ i ∈ s, f i a",
"usedConstants": [
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : Set α\nh : count s = 0\nx : α\nhx : x ∈ s\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β → α\nhf : Function.Injective f\ns : Finset β\ns_mble : MeasurableSet ↑s\nfs_mble : MeasurableSet (f '' ↑s)\n⊢ MeasurableSet ↑(Finset.image f s)",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 778,
"column": 4
} | {
"line": 779,
"column": 15
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nf g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n ↦ ((fun n ↦ SimpleFunc.piecewise s hs (hf.ap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 780,
"column": 4
} | {
"line": 781,
"column": 15
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nf g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nx : α\nhx : x ∉ s\n⊢ Tendsto (fun n ↦ ((fun n ↦ SimpleFunc.piecewise s hs (hf.ap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 802,
"column": 4
} | {
"line": 802,
"column": 84
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : α → γ\ng' : γ → β\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\ninst✝ : TopologicalSpace β\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\ny : α\n⊢ Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).extend ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 83,
"column": 13
} | {
"line": 83,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\na : α\nh : dirac a = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 81
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : α → γ\ng' : γ → β\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\ninst✝ : TopologicalSpace β\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : γ\nhx : ¬∃ y, g y = x\n⊢ Tendsto (fun n ↦ ((fun n ↦ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 865,
"column": 2
} | {
"line": 865,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : AddZeroClass β\ninst✝ : TopologicalSpace β\nP : (f : α → β) → StronglyMeasurable f → Prop\nind : ∀ (c : β) ⦃s : Set α⦄ (hs : MeasurableSet s), P (s.indicator fun x ↦ c) ⋯\nadd :\n ∀ ⦃f g : α → β⦄ (hf : StronglyMeasurable f) (hg : Strongl... | induction s n using SimpleFunc.induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 137,
"column": 45
} | {
"line": 137,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Countable α\ninst✝ : MeasurableSingletonClass α\nμ : Measure α\n⊢ (sum fun a ↦ μ {a} • dirac a) = μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 943,
"column": 2
} | {
"line": 943,
"column": 37
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nm : MeasurableSpace α\nf g : α → E\ninst✝³ : TopologicalSpace E\ninst✝² : Preorder E\ninst✝¹ : OrderClosedTopology E\ninst✝ : PseudoMetrizableSpace E\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\n⊢ MeasurableSet {a | f a < g a}",
"usedConstants": [
"Eq.mpr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 247,
"column": 51
} | {
"line": 247,
"column": 62
} | [
{
"pp": "δ : Type u_3\nι : Type u_4\nmδ : MeasurableSpace δ\nc : ι → ℝ\nd : ι → δ\nh1 : ∀ (i : ι), 0 ≤ c i\nh2 : HasSum c 1\n⊢ HasSum (fun i ↦ (c i).toNNReal) 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 294,
"column": 6
} | {
"line": 295,
"column": 49
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : dirac x = dirac y\nA : Set α\nA_mble : MeasurableSet A\nobs : A.indicator 1 x = A.indicator 1 y\nx_in_A : x ∈ A\n⊢ x ∈ A ↔ y ∈ A",
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_iff",
"Membership.mem",
"id... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 296,
"column": 6
} | {
"line": 297,
"column": 91
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : dirac x = dirac y\nA : Set α\nA_mble : MeasurableSet A\nobs : A.indicator 1 x = A.indicator 1 y\nx_in_A : x ∉ A\n⊢ x ∈ A ↔ y ∈ A",
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"congrArg",
"Membership.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 40
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : ∀ (x_1 : Set α), MeasurableSet x_1 → x ∈ x_1 → y ∈ x_1\nA : Set α\nA_mble : MeasurableSet A\nx_in_A : x ∉ A\n⊢ x ∈ A ↔ y ∈ A",
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"congrArg",
"Membership.mem"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 188,
"column": 68
} | {
"line": 189,
"column": 37
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure ν\nthis✝ : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))\nf : ℕ × ℕ → Set α :=\n fun p ↦\n (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1\ne... | by
rw [this, Set.compl_union_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Perfect | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PerfectSpace α\nU : Set α\nhU : IsOpen[inst✝¹] U\n⊢ Preperfect U",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 218,
"column": 40
} | {
"line": 221,
"column": 28
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nhs : MeasurableSet s\nhs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ\nhνs : ν s ≠ 0\nthis : ¬SigmaFinite (μ.restrict s)\n⊢ μ s = ∞",
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
... | by
by_contra h
have h_lt_top : Fact (μ s < ∞) := ⟨Ne.lt_top h⟩
exact this inferInstance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1100,
"column": 6
} | {
"line": 1100,
"column": 21
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞\nh_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n ↦ sup... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 42
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ ∫⁻ (x : α), c ∂μ < ∞",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → ℝ≥0∞\nf_bdd : ∃ c, ∀ (x : α), f x ≤ ↑c\n⊢ ∃ y, ∀ x ∈ univ, f x ≤ ↑y",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nμ : Measure α\ninst✝ : SFinite μ\nh : IsFiniteMeasure μ\nn : ℕ\n⊢ ∃ g, Measurable g ∧ g ≤ f ∧ g ≤ ↑n ∧ ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g a ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ∞\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.FunctionSeries | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 13
} | [
{
"pp": "β : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nι : Type u_4\nf : ι → β → F\nu : ι → ℝ\nhu : Summable u\nhfu : ∀ᶠ (n : ι) in cofinite, ∀ (x : β), ‖f n x‖ ≤ u n\n⊢ ∀ᶠ (n : ι) in cofinite, ∀ x ∈ univ, ‖f n x‖ ≤ u n",
"usedConstants": [
"Norm.norm",
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 46
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε\nx y : Y\nz : X\n⊢ glueDist Φ Ψ ε (Sum.inr x) (Sum.inl z) ≤\n glueDist Φ Ψ ε (Sum.inr x) (Sum.inr y) + g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 147,
"column": 4
} | {
"line": 148,
"column": 11
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε\nx y : X\nz : Y\n⊢ glueDist Φ Ψ ε (Sum.inl x) (Sum.inr z) ≤\n glueDist Φ Ψ ε (Sum.inl x) (Sum.inl y) + g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 25
} | [
{
"pp": "case mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhy : y ∈ cylinder x n\n⊢ cylinder y n = cylinder x n",
"usedConstants": [
"Set.Subset.antisymm",
"PiNat.cylinder",
"Nat"
]
}
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 23
} | [
{
"pp": "case h.mp\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nk : E n\nhk : ∀ i < n + 1, y i = update x n k i\ni : ℕ\nhi : i < n\n⊢ y i = x i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Polish | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsOpen[inst✝¹] s\n⊢ IsClopenable s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 361,
"column": 6
} | {
"line": 361,
"column": 17
} | [
{
"pp": "case inl.inl\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx z y : E i\n⊢ dist ⟨i, x⟩ ⟨i, z⟩ ≤ dist ⟨i, x⟩ ⟨i, y⟩ + dist ⟨i, y⟩ ⟨i, z⟩",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"id",
"LE.le",
"Real.inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 365,
"column": 10
} | {
"line": 365,
"column": 47
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx : E i\nj : ι\ny : E j\nz : E i\nhij : i ≠ j\n⊢ dist x z ≤ dist x ⋯.some + 0 + 0 + (0 + 0 + dist ⋯.some z)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"Real.instAddM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 429,
"column": 4
} | {
"line": 433,
"column": 45
} | [
{
"pp": "case refine_3.inr\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx : E i\nj : ι\ny : E j\nhij : i ≠ j\n⊢ Sigma.dist ⟨i, x⟩ ⟨j, y⟩ = 0 → ⟨i, x⟩ = ⟨j, y⟩",
"usedConstants": [
"Real.instLE",
"Real",
"Trans.trans",
"Preorder.toLT",
"lt_irrefl"... | · intro h
apply (lt_irrefl (1 : ℝ) _).elim
calc
1 ≤ Sigma.dist (⟨i, x⟩ : Σ k, E k) ⟨j, y⟩ := Sigma.one_le_dist_of_ne hij _ _
_ < 1 := by rw [h]; exact zero_lt_one | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 450,
"column": 4
} | {
"line": 451,
"column": 60
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝¹ : (i : ι) → MetricSpace (E i)\ninst✝ : ∀ (i : ι), CompleteSpace (E i)\ns : ι → Set ((i : ι) × E i) := fun i ↦ Sigma.fst ⁻¹' {i}\nU : Set (((k : ι) × E k) × (k : ι) × E k) := {p | dist p.1 p.2 < 1}\ni : ι\n⊢ IsComplete (s i)",
"usedConstants": [
"Eq.mpr",... | simp only [s, ← range_sigmaMk]
exact (isometry_mk i).isUniformInducing.isComplete_range | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 450,
"column": 4
} | {
"line": 451,
"column": 60
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝¹ : (i : ι) → MetricSpace (E i)\ninst✝ : ∀ (i : ι), CompleteSpace (E i)\ns : ι → Set ((i : ι) × E i) := fun i ↦ Sigma.fst ⁻¹' {i}\nU : Set (((k : ι) × E k) × (k : ι) × E k) := {p | dist p.1 p.2 < 1}\ni : ι\n⊢ IsComplete (s i)",
"usedConstants": [
"Eq.mpr",... | simp only [s, ← range_sigmaMk]
exact (isometry_mk i).isUniformInducing.isComplete_range | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 83
} | [
{
"pp": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : dist x y < (1 / 2) ^ n\ni : ℕ\nhi : i ≤ n\nhne : x ≠ y\n⊢ n < firstDiff x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 564,
"column": 35
} | {
"line": 564,
"column": 50
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx y : (n : ℕ) × X n\nm : ℕ\nhx : x.fst ≤ m + 1\nhy : y.fst ≤ m + 1\nh : ¬max x.fst y.fst = m + 1\nthis : max x.fst y.fst ≤ m.succ\n⊢ max x.fst y.fst ≤ m",
"usedConstants": [
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 333,
"column": 2
} | {
"line": 339,
"column": 36
} | [
{
"pp": "case mpr\nE : ℕ → Type u_1\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nf : ((n : ℕ) → E n) → α\n⊢ (∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n) →\n ∀ (x y : (n : ℕ) → E n), dist (f x) (f y) ≤ dist x y",
"usedConstants": [
"Eq.mpr",
"Real.instLE"... | · intro H x y
rcases eq_or_ne x y with (rfl | hne)
· simp [PiNat.dist_nonneg]
rw [dist_eq_of_ne hne]
apply H x y (firstDiff x y)
rw [firstDiff_comm]
exact mem_cylinder_firstDiff _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 574,
"column": 20
} | {
"line": 574,
"column": 45
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx : (n : ℕ) × X n\n⊢ inductiveLimitDist f x x = 0",
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Sigma.fst",
"LinearOrder.toMax",
"dist_... | simp [inductiveLimitDist] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 574,
"column": 20
} | {
"line": 574,
"column": 45
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx : (n : ℕ) × X n\n⊢ inductiveLimitDist f x x = 0",
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Sigma.fst",
"LinearOrder.toMax",
"dist_... | simp [inductiveLimitDist] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 574,
"column": 20
} | {
"line": 574,
"column": 45
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx : (n : ℕ) × X n\n⊢ inductiveLimitDist f x x = 0",
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Sigma.fst",
"LinearOrder.toMax",
"dist_... | simp [inductiveLimitDist] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 61,
"column": 96
} | {
"line": 66,
"column": 57
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α✝\nmβ : MeasurableSpace β\nα : Type u_3\nm : MeasurableSpace α\n⊢ MeasurableAdd₂ (Measure α)",
"usedConstants": [
"ENNReal.instAdd",
"MeasureTheory.Measure.instMeasurableSpace",
"MeasureTheory.Measure",
"MeasurableSet",
... | by
refine ⟨Measure.measurable_of_measurable_coe _ fun s hs => ?_⟩
simp_rw [Measure.coe_add, Pi.add_apply]
refine Measurable.add ?_ ?_
· exact (Measure.measurable_coe hs).comp measurable_fst
· exact (Measure.measurable_coe hs).comp measurable_snd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 45
} | [
{
"pp": "case iUnion\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : α → Measure β\ninst✝ : ∀ (a : α), IsFiniteMeasure (μ a)\nS : Set (Set β)\nhgen : mβ = MeasurableSpace.generateFrom S\nhpi : IsPiSystem S\nh_basic : ∀ s ∈ S, Measurable fun a ↦ (μ a) s\nh_univ : Measurable fun a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 36
} | [
{
"pp": "case mpr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nh : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (n : ℕ) → E n), dist x y < ε → y ∈ s\nx : (n : ℕ) → E n\nhx : x ∈ s\n⊢ ∃ t ∈ {s | ∃ x n, s = cylinder x n}, x ∈ t ∧ t ⊆ s",
"usedCo... | rcases h x hx with ⟨ε, εpos, hε⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 38
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 74
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nm : MeasurableSpace X\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nhU : IsOpen[inst✝¹] U\n⊢ μ U = 0 ↔ U = ∅",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 44
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\nm : MeasurableSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nf g : X → Y\nhU : IsOpen[inst✝³] U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : ∀ᵐ (x : X) ∂μ, x ∈ U → f x = g x\n... | simp only [ae_iff, Classical.not_imp] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 25
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : Countable ι\ninst✝ : T2Space α\ns : ι → Set α\nhs : ∀ (n : ι), AnalyticSet (s n)\ni₀ : ι\nβ : ι → Type\nhβ : (n : ι) → TopologicalSpace (β n)\nh'β : ∀ (n : ι), PolishSpace (β n)\nf : (n : ι) → β n → α\nf_cont : ∀ (n : ι), Continuous[hβ n... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 531,
"column": 4
} | {
"line": 531,
"column": 58
} | [
{
"pp": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∉ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\n⊢ ∃ y ∈ s, x ∈ cyl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 269,
"column": 8
} | {
"line": 269,
"column": 35
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : Countable ι\ninst✝ : T2Space α\ns : ι → Set α\nhs : ∀ (n : ι), AnalyticSet (s n)\ni₀ : ι\nβ : ι → Type\nhβ : (n : ι) → TopologicalSpace (β n)\nh'β : ∀ (n : ι), PolishSpace (β n)\nf : (n : ι) → β n → α\nf_cont : ∀ (n : ι), Continuous[hβ n... | rw [← mem_range, f_range n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 593,
"column": 4
} | {
"line": 593,
"column": 25
} | [
{
"pp": "case refine_1\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\n⊢ ra... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 597,
"column": 6
} | {
"line": 597,
"column": 32
} | [
{
"pp": "case neg\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\ny : (n : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Convolution | {
"line": 47,
"column": 61
} | {
"line": 48,
"column": 45
} | [
{
"pp": "M : Type u_1\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nμ ν : Measure M\nf : M → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (z : M), f z ∂μ ∗ₘ ν = ∫⁻ (z : M × M), f (z.1 * z.2) ∂μ.prod ν",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"... | by
rw [mconv, lintegral_map hf measurable_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 656,
"column": 14
} | {
"line": 656,
"column": 90
} | [
{
"pp": "case inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n ... | exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 79,
"column": 14
} | {
"line": 80,
"column": 11
} | [
{
"pp": "case basic\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : IsFiniteMeasure ν\ns✝ : Set (α × β)\ns : Set α\nhs : s ∈ {s | MeasurableSet s}\nt : Set β\n⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' (fun x1 x2 ↦ x1 ×ˢ x2) s t)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 27
} | [
{
"pp": "case iUnion\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : IsFiniteMeasure ν\ns : Set (α × β)\nf : ℕ → Set (α × β)\nhfd : Pairwise (Disjoint on f)\nhfm : ∀ (i : ℕ), MeasurableSet (f i)\nihf : ∀ (i : ℕ), Measurable fun x ↦ ν (Prod.mk x ⁻¹' f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 25
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' s)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Measurab... | rw [← sum_sfiniteSeq ν] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 779,
"column": 4
} | {
"line": 779,
"column": 40
} | [
{
"pp": "α : Type u_2\ninst✝³ : MetricSpace α\ninst✝² : CompleteSpace α\ninst✝¹ : SecondCountableTopology α\ninst✝ : Nonempty α\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : ℕ → α\nhu : DenseRange u\ns : Set (ℕ → ℕ) := ⋯\ng : ↑s → α := ⋯\nA : ∀ (x : ↑s) (n : ℕ), dist (g x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 41
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → EDist (F i)\nx y : (i : ι) → F i\ni : ι\nh : edist x y < 2⁻¹ ^ encode i\n⊢ edist (x i) (y i) ≤ edist x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 853,
"column": 18
} | {
"line": 853,
"column": 29
} | [
{
"pp": "E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ ε / 2 > 0",
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"instHDiv",
"and_true",
"congrArg",
"PartialOrder.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 855,
"column": 48
} | {
"line": 855,
"column": 59
} | [
{
"pp": "E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\n⊢ ε / 2 ≠ 0",
"usedConstants": [
"Eq.mpr",
"False",
"instHDiv",
"con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 38
} | [
{
"pp": "case open_pos.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ : Measure γ\ninst✝⁵ : SFinite ν✝\nX : Type u_4\nY : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace... | exact v_open.measure_pos ν ⟨y, yv⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.