module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Normed.Group.Submodule | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 22
} | {
"line": 55,
"column": 2
} | [
{
"pp": "R : Type u_3\nR' : Type u_4\nM : Type u_5\nM' : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R'\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M\ninst✝² : Module R' M'\nσ₁₂ : R →+* R'\nf : M →ₛₗ[σ₁₂] M'\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalSpace M'\nhf : Continu... | [
"R : Type u_3\nR' : Type u_4\nM : Type u_5\nM' : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R'\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M\ninst✝² : Module R' M'\nσ₁₂ : R →+* R'\nf : M →ₛₗ[σ₁₂] M'\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalSpace M'\nhf : Continuous ⇑f\np : ... | rw [coe_domRestrict] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.AddTorsor | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 81
} | {
"line": 133,
"column": 2
} | [
{
"pp": "G : Type u_1\nP : Type u_2\ninst✝³ : LE G\ninst✝² : Preorder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\na b : G\nc d : P\nh₁ : a ≤ b\nh₂ : c < d\n⊢ b • c < b • d",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"instHSMul",
"le_of_lt",
"Preorder.toLE",... | [
"G : Type u_1\nP : Type u_2\ninst✝³ : LE G\ninst✝² : Preorder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\na b : G\nc d : P\nh₁ : a ≤ b\nh₂ : c < d\n⊢ ¬b • d ≤ b • c"
] | refine lt_of_le_not_ge (IsOrderedSMul.smul_le_smul_left c d (le_of_lt h₂) b) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.MulAction | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 46
} | {
"line": 180,
"column": 6
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : MulActionWithZero α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\nh : ¬r = 0\n⊢ ‖r • x‖ = ‖r‖ * ‖x‖",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"norm_smul_le",
... | [
"case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : MulActionWithZero α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\nh : ¬r = 0\n⊢ ‖r‖ * ‖x‖ ≤ ‖r • x‖"
] | refine le_antisymm (norm_smul_le r x) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 984,
"column": 4
} | {
"line": 984,
"column": 73
} | {
"line": 985,
"column": 4
} | [
{
"pp": "case inr.refine_2\nα : Type u_5\ninst✝¹ : Fintype α\nι : α → Type u\ninst✝ : ∀ (a : α), Finite (ι a)\nf : (a : α) → ι a → ℝ\nhf₀ : ∀ (a : α) (i : ι a), 0 ≤ f a i\nh : ∀ (i : α), Nonempty (ι i)\n⊢ ∏ a, ⨆ i, f a i ≤ ⨆ i, ∏ a, f a (i a)",
"ppTerm": "?inr.refine_2",
"assigned": true,
"usedConst... | [
"case inr.refine_2\nα : Type u_5\ninst✝¹ : Fintype α\nι : α → Type u\ninst✝ : ∀ (a : α), Finite (ι a)\nf : (a : α) → ι a → ℝ\nhf₀ : ∀ (a : α) (i : ι a), 0 ≤ f a i\nh : ∀ (i : α), Nonempty (ι i)\nH : ∀ (a : α), ∃ i, f a i = ⨆ i, f a i\n⊢ ∏ a, ⨆ i, f a i ≤ ⨆ i, ∏ a, f a (i a)"
] | have H a : ∃ i : ι a, f a i = ⨆ i, f a i := exists_eq_ciSup_of_finite | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Prod.TProd | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 18
} | {
"line": 111,
"column": 4
} | [
{
"pp": "ι : Type u\nα : ι → Type v\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nhl : (i :: is).Nodup\nv w : TProd α (i :: is)\nhvw : ∀ (i_1 : ι) (hi : i_1 ∈ i :: is), v.elim hi = w.elim hi\n⊢ v = w",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"PUnit",
"Prod.ext",
"Prod... | [
"case fst\nι : Type u\nα : ι → Type v\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nhl : (i :: is).Nodup\nv w : TProd α (i :: is)\nhvw : ∀ (i_1 : ι) (hi : i_1 ∈ i :: is), v.elim hi = w.elim hi\n⊢ v.1 = w.1",
"case snd\nι : Type u\nα : ι → Type v\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nhl : (i :: is).Nodup\nv w ... | apply Prod.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Prod.TProd | {
"line": 111,
"column": 10
} | {
"line": 111,
"column": 24
} | {
"line": 111,
"column": 25
} | [
{
"pp": "case fst\nι : Type u\nα : ι → Type v\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nhl : (i :: is).Nodup\nv w : TProd α (i :: is)\nhvw : ∀ (i_1 : ι) (hi : i_1 ∈ i :: is), v.elim hi = w.elim hi\n⊢ v.1 = w.1",
"ppTerm": "?fst",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.TPr... | [
"case fst\nι : Type u\nα : ι → Type v\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nhl : (i :: is).Nodup\nv w : TProd α (i :: is)\nhvw : ∀ (i_1 : ι) (hi : i_1 ∈ i :: is), v.elim hi = w.elim hi\n⊢ v.elim ⋯ = w.1"
] | ← elim_self v, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 174,
"column": 6
} | {
"line": 175,
"column": 11
} | {
"line": 176,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedDivisionRing α\nx : α\nhx : x ≠ 0\n⊢ (closedBall x (‖x‖ / 2))ᶜ ∈ 𝓝 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNor... | [] | refine Metric.isClosed_closedBall.isOpen_compl.mem_nhds ?_
simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 174,
"column": 6
} | {
"line": 175,
"column": 11
} | {
"line": 176,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedDivisionRing α\nx : α\nhx : x ≠ 0\n⊢ (closedBall x (‖x‖ / 2))ᶜ ∈ 𝓝 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNor... | [] | refine Metric.isClosed_closedBall.isOpen_compl.mem_nhds ?_
simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 288,
"column": 4
} | {
"line": 289,
"column": 65
} | {
"line": 290,
"column": 2
} | [
{
"pp": "case mpr.inl\nK : Type u_4\ninst✝ : NormedField K\nh : IsComplete (closedBall 0 1)\nh✝ : DiscreteTopology K\n⊢ CompleteSpace K",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"IsComplete",
"C... | [] | rwa [completeSpace_iff_isComplete_univ,
← NormedDivisionRing.unitClosedBall_eq_univ_of_discrete] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 288,
"column": 4
} | {
"line": 289,
"column": 65
} | {
"line": 290,
"column": 2
} | [
{
"pp": "case mpr.inl\nK : Type u_4\ninst✝ : NormedField K\nh : IsComplete (closedBall 0 1)\nh✝ : DiscreteTopology K\n⊢ CompleteSpace K",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"IsComplete",
"C... | [] | rwa [completeSpace_iff_isComplete_univ,
← NormedDivisionRing.unitClosedBall_eq_univ_of_discrete] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 288,
"column": 4
} | {
"line": 289,
"column": 65
} | {
"line": 290,
"column": 2
} | [
{
"pp": "case mpr.inl\nK : Type u_4\ninst✝ : NormedField K\nh : IsComplete (closedBall 0 1)\nh✝ : DiscreteTopology K\n⊢ CompleteSpace K",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"IsComplete",
"C... | [] | rwa [completeSpace_iff_isComplete_univ,
← NormedDivisionRing.unitClosedBall_eq_univ_of_discrete] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 48
} | {
"line": 240,
"column": 4
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : Infinite 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : Nontrivial E\ninst✝ : NormedSpace 𝕜 E\nH : ∃ c, c ≠ 0 ∧ ‖c‖ ≠ 1\n⊢ NoncompactSpace E",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NontriviallyNo... | [
"case pos\n𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : Infinite 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : Nontrivial E\ninst✝ : NormedSpace 𝕜 E\nH : ∃ c, c ≠ 0 ∧ ‖c‖ ≠ 1\nthis : NontriviallyNormedField 𝕜 := NontriviallyNormedField.ofNormNeOne H\n⊢ NoncompactSpace E"
] | let := NontriviallyNormedField.ofNormNeOne H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 333,
"column": 6
} | {
"line": 333,
"column": 34
} | {
"line": 334,
"column": 2
} | [
{
"pp": "case h₂\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\nz : β\nhz : z ∈ (measurableAtom x)ᶜ\n⊢ ∃ i ∉ measurableAtom x, z ∉ s i",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
... | [] | exact ⟨z, hz, (hs z hz).2.2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 85,
"column": 30
} | {
"line": 85,
"column": 52
} | {
"line": 85,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm✝ : OuterMeasure α\nR : Type u_3\ninst✝³ : SMul R ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nR' : Type u_4\ninst✝¹ : SMul R' ℝ≥0∞\ninst✝ : IsScalarTower R' ℝ≥0∞ ℝ≥0∞\nc : R\nm : OuterMeasure α\ns t : Set α\nh : s ⊆ t\n⊢ c • 1 * m s ≤ c • m t",
"ppTerm": "?m.47",
"ass... | [
"α : Type u_1\nβ : Type u_2\nm✝ : OuterMeasure α\nR : Type u_3\ninst✝³ : SMul R ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nR' : Type u_4\ninst✝¹ : SMul R' ℝ≥0∞\ninst✝ : IsScalarTower R' ℝ≥0∞ ℝ≥0∞\nc : R\nm : OuterMeasure α\ns t : Set α\nh : s ⊆ t\n⊢ c • 1 * m s ≤ c • 1 * m t"
] | ← smul_one_mul c (m t) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 26
} | {
"line": 530,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort uι\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : Countable ι\nt : ι → Set α\nf : (i : ι) → ↑(t i) → β\nhtf : ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩\nT : Set α\nhT : T ⊆ ⋃ i, t i\nhtm : ∀ (i : ι), MeasurableSet (t i... | [
"α : Type u_1\nβ : Type u_2\nι : Sort uι\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : Countable ι\nt : ι → Set α\nf : (i : ι) → ↑(t i) → β\nhtf : ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩\nT : Set α\nhT : T ⊆ ⋃ i, t i\nhtm : ∀ (i : ι), MeasurableSet (t i)\nhfm : ∀ (... | rw [preimage_iUnionLift] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 1008,
"column": 44
} | {
"line": 1008,
"column": 62
} | {
"line": 1008,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : MeasurableSpace β\ng : β → Finset α\n⊢ (Measurable fun x ↦ ↑(g x)) ↔ ∀ (a : α), Measurable fun x ↦ a ∈ g x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Measurable",
"Membership.me... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : MeasurableSpace β\ng : β → Finset α\n⊢ (∀ (a : α), Measurable fun x ↦ a ∈ ↑(g x)) ↔ ∀ (a : α), Measurable fun x ↦ a ∈ g x"
] | measurable_set_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 8
} | {
"line": 68,
"column": 8
} | [
{
"pp": "β : Type u_2\nf : β → ℝ≥0\na : ℝ≥0\nha : ↑a = ∑' (b : β), ↑(f b)\n⊢ HasSum (fun a ↦ ↑(f a)) ↑a",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"congrArg",
"id",
"tsum",
"HasSum",
... | [
"β : Type u_2\nf : β → ℝ≥0\na : ℝ≥0\nha : ↑a = ∑' (b : β), ↑(f b)\n⊢ HasSum (fun a ↦ ↑(f a)) (∑' (b : β), ↑(f b))"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 281,
"column": 6
} | {
"line": 281,
"column": 59
} | {
"line": 281,
"column": 59
} | [
{
"pp": "α : Type u_1\nι : Type u_4\ninst✝ : Fintype ι\nf : α → ℝ≥0∞\nt : ι → Set α\n⊢ ∑' (x : ↑(⋃ i, t i)), f ↑x ≤ ∑ i, ∑' (x : ↑(t i)), f ↑x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"ENNReal.instAddCommMonoid",
"congrArg",
"Me... | [
"α : Type u_1\nι : Type u_4\ninst✝ : Fintype ι\nf : α → ℝ≥0∞\nt : ι → Set α\n⊢ ∑' (x : ↑(⋃ i, t i)), f ↑x ≤ ∑' (b : ι) (x : ↑(t b)), f ↑x"
] | ← tsum_fintype (L := SummationFilter.unconditional _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 31
} | {
"line": 305,
"column": 2
} | [
{
"pp": "ι : Type u_4\na : ι → ℝ≥0∞\ntsum_ne_top : ∑' (i : ι), a i ≠ ∞\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nh : ¬{i | ε ≤ a i}.Finite\n⊢ False",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ofPred",
"Set.Elem",
"LE.le",
"ENNReal.instLE",
"ENNReal",
"Infin... | [
"ι : Type u_4\na : ι → ℝ≥0∞\ntsum_ne_top : ∑' (i : ι), a i ≠ ∞\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nh : ¬{i | ε ≤ a i}.Finite\nthis : Infinite ↑{i | ε ≤ a i}\n⊢ False"
] | have := Infinite.to_subtype h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.EReal.Inv | {
"line": 242,
"column": 23
} | {
"line": 242,
"column": 85
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case pos_bot\nx : ℝ\nx_pos : 0 < x\n⊢ (↑x * ⊥)⁻¹ = (↑x)⁻¹ * ⊥⁻¹",
"ppTerm": "?pos_bot",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real",
"Preorder.toLT",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"MulZeroClass.toMul",
"Real.in... | [] | rw [mul_bot_of_pos (EReal.coe_pos.2 x_pos), inv_bot, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Inv | {
"line": 242,
"column": 23
} | {
"line": 242,
"column": 85
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case pos_bot\nx : ℝ\nx_pos : 0 < x\n⊢ (↑x * ⊥)⁻¹ = (↑x)⁻¹ * ⊥⁻¹",
"ppTerm": "?pos_bot",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real",
"Preorder.toLT",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"MulZeroClass.toMul",
"Real.in... | [] | rw [mul_bot_of_pos (EReal.coe_pos.2 x_pos), inv_bot, mul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Inv | {
"line": 242,
"column": 23
} | {
"line": 242,
"column": 85
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case pos_bot\nx : ℝ\nx_pos : 0 < x\n⊢ (↑x * ⊥)⁻¹ = (↑x)⁻¹ * ⊥⁻¹",
"ppTerm": "?pos_bot",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real",
"Preorder.toLT",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"MulZeroClass.toMul",
"Real.in... | [] | rw [mul_bot_of_pos (EReal.coe_pos.2 x_pos), inv_bot, mul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Inv | {
"line": 244,
"column": 27
} | {
"line": 244,
"column": 67
} | {
"line": 244,
"column": 68
} | [
{
"pp": "case neg_bot\nx : ℝ\nx_neg : x < 0\n⊢ (↑x * ⊥)⁻¹ = (↑x)⁻¹ * ⊥⁻¹",
"ppTerm": "?neg_bot",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real",
"Preorder.toLT",
"HMul.hMul",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
... | [
"case neg_bot\nx : ℝ\nx_neg : x < 0\n⊢ ⊤⁻¹ = (↑x)⁻¹ * ⊥⁻¹"
] | mul_bot_of_neg (EReal.coe_neg'.2 x_neg), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 55
} | {
"line": 318,
"column": 4
} | [
{
"pp": "b✝ : EReal\na : ℝ\na_0 : 0 < ↑a\nb : ℝ\nb_0 : 0 < ↑b\na_b : ↑a < ↑b\n⊢ (↑b)⁻¹ < (↑a)⁻¹",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"Real.instInv",
"PartialOrder.toPreorder",
"EReal",
... | [
"b✝ : EReal\na : ℝ\na_0 : 0 < ↑a\nb : ℝ\nb_0 : 0 < ↑b\na_b : ↑a < ↑b\n⊢ b⁻¹ < a⁻¹"
] | rw [← coe_inv a, ← coe_inv b, EReal.coe_lt_coe_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 635,
"column": 2
} | {
"line": 635,
"column": 50
} | {
"line": 636,
"column": 2
} | [
{
"pp": "α : Type u_4\ns : Set α\n⊢ ∑' (x : ↑s), 1 = ↑s.encard",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.encard",
"ENNReal.instAddCommMonoid",
"Set.finite_or_infinite",
"Set.Finite",
"Set.Elem",
"ENat.toENNReal",
"Or.casesOn",
"tsu... | [
"case inl\nα : Type u_4\ns : Set α\nhfin : s.Finite\n⊢ ∑' (x : ↑s), 1 = ↑s.encard",
"case inr\nα : Type u_4\ns : Set α\nhinf : s.Infinite\n⊢ ∑' (x : ↑s), 1 = ↑s.encard"
] | obtain (hfin | hinf) := Set.finite_or_infinite s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 344,
"column": 15
} | {
"line": 344,
"column": 23
} | {
"line": 344,
"column": 24
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\nγ : Type u_4\ninst✝² : LinearOrder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : ClosedIciTopology γ\nf : α → γ\ns : Set α\nhs : IsClosed[inst✝³] s\nhf : ∀ x ∈ s, ∀ y < f x, ∀ᶠ (x : α) in 𝓝 x, x ∈ s → y < f x\nx : α\ny : γ\nh : (x, y).1 ∈ s → (x, y).2 < f (x, y).1\... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\nγ : Type u_4\ninst✝² : LinearOrder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : ClosedIciTopology γ\nf : α → γ\ns : Set α\nhs : IsClosed[inst✝³] s\nhf : ∀ x ∈ s, ∀ y < f x, ∀ᶠ (x : α) in 𝓝 x, x ∈ s → y < f x\nx : α\ny : γ\nh : (x, y).1 ∈ s → (x, y).2 < f (x, y).1\nhx : x ∈ s\... | ⟨h₂, h₃⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 374,
"column": 37
} | {
"line": 374,
"column": 39
} | {
"line": 375,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_4\ninst✝⁵ : LinearOrder γ\ninst✝⁴ : TopologicalSpace γ\ninst✝³ : OrderTopology γ\nδ : Type u_5\ninst✝² : LinearOrder δ\ninst✝¹ : TopologicalSpace δ\ninst✝ : OrderTopology δ\ng : γ → δ\nf : α → γ\nhg : ContinuousAt g (f x)\nhf : Low... | [
"α : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_4\ninst✝⁵ : LinearOrder γ\ninst✝⁴ : TopologicalSpace γ\ninst✝³ : OrderTopology γ\nδ : Type u_5\ninst✝² : LinearOrder δ\ninst✝¹ : TopologicalSpace δ\ninst✝ : OrderTopology δ\ng : γ → δ\nf : α → γ\nhg : ContinuousAt g (f x)\nhf : LowerSemicontin... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 71
} | {
"line": 271,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\ns : Set α\nm_empty : m ∅ = 0\nm_mono : ∀ ⦃t : Set α⦄, s ⊆ t → m s ≤ m t\nm_subadd : ∀ (s : ℕ → Set α), m (⋃ i, s i) ≤ ∑' (i : ℕ), m (s i)\n⊢ (boundedBy m) s = m s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.O... | [] | rw [boundedBy_eq_ofFunction m_empty, ofFunction_eq s m_mono m_subadd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 71
} | {
"line": 271,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\ns : Set α\nm_empty : m ∅ = 0\nm_mono : ∀ ⦃t : Set α⦄, s ⊆ t → m s ≤ m t\nm_subadd : ∀ (s : ℕ → Set α), m (⋃ i, s i) ≤ ∑' (i : ℕ), m (s i)\n⊢ (boundedBy m) s = m s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.O... | [] | rw [boundedBy_eq_ofFunction m_empty, ofFunction_eq s m_mono m_subadd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 71
} | {
"line": 271,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\ns : Set α\nm_empty : m ∅ = 0\nm_mono : ∀ ⦃t : Set α⦄, s ⊆ t → m s ≤ m t\nm_subadd : ∀ (s : ℕ → Set α), m (⋃ i, s i) ≤ ∑' (i : ℕ), m (s i)\n⊢ (boundedBy m) s = m s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.O... | [] | rw [boundedBy_eq_ofFunction m_empty, ofFunction_eq s m_mono m_subadd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SetDissipate | {
"line": 31,
"column": 94
} | {
"line": 32,
"column": 41
} | {
"line": 34,
"column": 0
} | [
{
"pp": "β : Type u_2\ns : ℕ → Set β\nn : ℕ\n⊢ dissipate s n = ⋂ k, ⋂ (_ : k < n + 1), s k",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Set.dissipate",
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"Set.iInter",
"_private.Mathlib.Order.SetDissipate.0.Set.dissip... | [] | by
simp_rw [Nat.lt_add_one_iff, dissipate] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 739,
"column": 86
} | {
"line": 740,
"column": 62
} | {
"line": 742,
"column": 0
} | [
{
"pp": "R : Type u_4\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\n⊢ Tendsto (fun x ↦ ↑⌊x⌋₊ / x) atTop (𝓝 1)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"instHDiv",
"H... | [] | by
simpa using tendsto_nat_floor_mul_div_atTop (zero_le_one' R) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 22
} | {
"line": 167,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ μ s = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), μ t",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"iInf",
"MeasurableSet",
"congrArg",
"... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ μ.trim s = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), μ t"
] | measure_eq_trim, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.OuterMeasure.AE | {
"line": 103,
"column": 29
} | {
"line": 103,
"column": 31
} | {
"line": 103,
"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\nι : Sort u_4\np : α → ι → Prop\nhp : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i\ni : ι\na : α\n⊢ (∀ (i : ι), p a i) → p a i",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [
"α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\nι : Sort u_4\np : α → ι → Prop\nhp : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i\ni : ι\na : α\nha : ∀ (i : ι), p a i\n⊢ p a i"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.PiSystem | {
"line": 706,
"column": 2
} | {
"line": 706,
"column": 17
} | {
"line": 706,
"column": 18
} | [
{
"pp": "case basic\nα : Type u_3\nm : MeasurableSpace α\nC : (s : Set α) → MeasurableSet s → Prop\ns : Set (Set α)\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nempty : C ∅ ⋯\nbasic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯\ncompl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯\niUnion :\n ∀ (f : ℕ ... | [] | | basic u hu => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 53
} | {
"line": 133,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort uι\ninst✝¹ : MeasurableSpace α\nf : ι → Filter α\ninst✝ : ∀ (i : ι), (f i).IsMeasurablyGenerated\nt : Set (PLift ι)\nht : t.Finite\nV : ↑t → Set α\nhVf : ∀ (i : ↑t), V i ∈ f (Equiv.plift ↑i)\nU : ↑t → Set α\nhUf : ∀ (i : ↑t... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort uι\ninst✝¹ : MeasurableSpace α\nf : ι → Filter α\ninst✝ : ∀ (i : ι), (f i).IsMeasurablyGenerated\nt : Set (PLift ι)\nht : t.Finite\nV : ↑t → Set α\nhVf : ∀ (i : ↑t), V i ∈ f (Equiv.plift ↑i)\nU : ↑t → Set α\nhUf : ∀ (i : ↑t), U i ∈ f (... | rw [← Equiv.plift.surjective.iInf_comp, mem_iInf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 29
} | {
"line": 202,
"column": 2
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : NullMeasurableSet s μ\n⊢ ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"MeasurableSet",
"MeasureTheory.Nu... | [
"α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhtm : MeasurableSet t\nhst : s =ᵐ[μ] t\n⊢ ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s"
] | rcases h with ⟨t, htm, hst⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 306,
"column": 30
} | {
"line": 306,
"column": 96
} | {
"line": 308,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : NullMeasurableSet s μ\n⊢ μ s + μ sᶜ = μ univ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"MeasureTheory.Measure",
"MeasureTheory.aedisjoint_compl_right",
... | [] | rw [← measure_union₀' hs aedisjoint_compl_right, union_compl_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 306,
"column": 30
} | {
"line": 306,
"column": 96
} | {
"line": 308,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : NullMeasurableSet s μ\n⊢ μ s + μ sᶜ = μ univ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"MeasureTheory.Measure",
"MeasureTheory.aedisjoint_compl_right",
... | [] | rw [← measure_union₀' hs aedisjoint_compl_right, union_compl_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 306,
"column": 30
} | {
"line": 306,
"column": 96
} | {
"line": 308,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : NullMeasurableSet s μ\n⊢ μ s + μ sᶜ = μ univ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"MeasureTheory.Measure",
"MeasureTheory.aedisjoint_compl_right",
... | [] | rw [← measure_union₀' hs aedisjoint_compl_right, union_compl_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 366,
"column": 4
} | {
"line": 366,
"column": 35
} | {
"line": 366,
"column": 35
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝² : MeasurableSingletonClass α\nmβ : MeasurableSpace β\ninst✝¹ : MeasurableSingletonClass β\ninst✝ : Countable β\nμ : Measure (α × β)\nx : α\n⊢ μ (Prod.fst ⁻¹' {x}) = μ (range (Prod.mk x))",
"ppTerm": "?m.47",
"assigned": true,
"usedC... | [
"α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝² : MeasurableSingletonClass α\nmβ : MeasurableSpace β\ninst✝¹ : MeasurableSingletonClass β\ninst✝ : Countable β\nμ : Measure (α × β)\nx : α\n⊢ μ (range fun x_1 ↦ (x, x_1)) = μ (range (Prod.mk x))"
] | preimage_fst_singleton_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 370,
"column": 59
} | {
"line": 370,
"column": 87
} | {
"line": 371,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝² : MeasurableSingletonClass α\nmβ : MeasurableSpace β\ninst✝¹ : MeasurableSingletonClass β\ninst✝ : Countable α\nμ : Measure (α × β)\ny✝ : β\ny : α × β\n⊢ y ∈ Prod.snd ⁻¹' {y✝} ↔ y ∈ ⋃ x, {(x, y✝)}",
"ppTerm": "?m.49",
"assigned": true,
... | [] | simp [Prod.ext_iff, eq_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Map | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 78
} | {
"line": 131,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nμ : Measure α\nf : α → β\n⊢ map f (c • μ) = c • map f μ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"instHSMul",
... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nμ : Measure α\nf : α → β\n⊢ ∀ (c : ℝ≥0∞), map f (c • μ) = c • map f μ"
] | suffices ∀ c : ℝ≥0∞, (c • μ).map f = c • μ.map f by simpa using this (c • 1) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.MeasureTheory.OuterMeasure.BorelCantelli | {
"line": 102,
"column": 30
} | {
"line": 103,
"column": 65
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns : ℕ → Set α\nh : ∑' (i : ℕ), μ (s i) ≠ ∞\n⊢ μ (liminf s atTop) = 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.liminf",
"MeasureTheory.measu... | [] | by
rw [← Nat.cofinite_eq_atTop, measure_liminf_cofinite_eq_zero h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 18
} | {
"line": 95,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf : α → β\nhf : QuasiMeasurePreserving f μa μb\nf' : α → β\nhf' : Measurable f'\nh : f =ᵐ[μa] f'\n⊢ QuasiMeasurePreserving f' μa μb",
"ppTerm": "?m.26",
"assigned": true,
"usedConstan... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf : α → β\nhf : QuasiMeasurePreserving f μa μb\nf' : α → β\nhf' : Measurable f'\nh : f =ᵐ[μa] f'\n⊢ map f' μa ≪ μb"
] | refine ⟨hf', ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 66
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → α\nh : QuasiMeasurePreserving f μ μ\nhs' : f ⁻¹' s =ᵐ[μ] s\nt : Set α\nhtm : MeasurableSet t\nht : s =ᵐ[μ] t\n⊢ MeasurableSet (limsup (fun x ↦ f^[x] ⁻¹' t) atTop)",
"ppTerm": "?refine_1",
"assigned": true,
... | [] | exact .measurableSet_limsup fun n ↦ h.measurable.iterate n htm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 66
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → α\nh : QuasiMeasurePreserving f μ μ\nhs' : f ⁻¹' s =ᵐ[μ] s\nt : Set α\nhtm : MeasurableSet t\nht : s =ᵐ[μ] t\n⊢ MeasurableSet (limsup (fun x ↦ f^[x] ⁻¹' t) atTop)",
"ppTerm": "?refine_1",
"assigned": true,
... | [] | exact .measurableSet_limsup fun n ↦ h.measurable.iterate n htm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 66
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → α\nh : QuasiMeasurePreserving f μ μ\nhs' : f ⁻¹' s =ᵐ[μ] s\nt : Set α\nhtm : MeasurableSet t\nht : s =ᵐ[μ] t\n⊢ MeasurableSet (limsup (fun x ↦ f^[x] ⁻¹' t) atTop)",
"ppTerm": "?refine_1",
"assigned": true,
... | [] | exact .measurableSet_limsup fun n ↦ h.measurable.iterate n htm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 257,
"column": 22
} | {
"line": 257,
"column": 97
} | {
"line": 257,
"column": 97
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx✝ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\ne : α ≃ᵐ β\n⊢ Measure.map (⇑e.symm) (Measure.map (⇑e) μ) ≪ μ",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"MeasureTheory.Me... | [] | by rw [Measure.map_map, e.symm_comp_self, Measure.map_id] <;> measurability | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 46,
"column": 16
} | {
"line": 46,
"column": 18
} | {
"line": 47,
"column": 6
} | [
{
"pp": "case pos.mp\nX : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\nb : X\nhsne : s.Nonempty\nthis : Nonempty ↑s\nhsub : s ⊆ Iio b\nh : comap Subtype.val (𝓝[<] b) = atTop\na : X\n⊢ a < b → (s ∩ Ioo a b).Nonempty",
"ppTerm": "?pos.mp✝",
"assigned"... | [
"case pos.mp\nX : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\nb : X\nhsne : s.Nonempty\nthis : Nonempty ↑s\nhsub : s ⊆ Iio b\nh : comap Subtype.val (𝓝[<] b) = atTop\na : X\nha : a < b\n⊢ (s ∩ Ioo a b).Nonempty"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 38,
"column": 70
} | {
"line": 38,
"column": 96
} | {
"line": 40,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\n⊢ ¬IsFiniteMeasure μ ↔ μ univ = ∞",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Measure.Typeclasses.Finite.0.MeasureTheory.not_isFiniteMeasure_iff._simp_1_1",
"MeasureTheory.Measure"... | [] | simp [isFiniteMeasure_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 38,
"column": 70
} | {
"line": 38,
"column": 96
} | {
"line": 40,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\n⊢ ¬IsFiniteMeasure μ ↔ μ univ = ∞",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Measure.Typeclasses.Finite.0.MeasureTheory.not_isFiniteMeasure_iff._simp_1_1",
"MeasureTheory.Measure"... | [] | simp [isFiniteMeasure_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 38,
"column": 70
} | {
"line": 38,
"column": 96
} | {
"line": 40,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\n⊢ ¬IsFiniteMeasure μ ↔ μ univ = ∞",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Measure.Typeclasses.Finite.0.MeasureTheory.not_isFiniteMeasure_iff._simp_1_1",
"MeasureTheory.Measure"... | [] | simp [isFiniteMeasure_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 62
} | {
"line": 220,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : MeasurableSet s\n⊢ (μ.restrict t).restrict s = (μ.restrict s).restrict t",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Mea... | [] | rw [restrict_restrict hs, restrict_restrict' hs, inter_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 62
} | {
"line": 220,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : MeasurableSet s\n⊢ (μ.restrict t).restrict s = (μ.restrict s).restrict t",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Mea... | [] | rw [restrict_restrict hs, restrict_restrict' hs, inter_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 62
} | {
"line": 220,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : MeasurableSet s\n⊢ (μ.restrict t).restrict s = (μ.restrict s).restrict t",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Mea... | [] | rw [restrict_restrict hs, restrict_restrict' hs, inter_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 324,
"column": 18
} | {
"line": 324,
"column": 36
} | {
"line": 326,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\nt : Set β\nht : MeasurableSet t\n⊢ ((map f μ).restrict s) t = (map f (μ.restrict (f ⁻¹' s))) t",
"ppTerm": "?m.36",
"assigned": true,
... | [] | by simp [*, hf ht] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 469,
"column": 2
} | {
"line": 469,
"column": 17
} | {
"line": 469,
"column": 18
} | [
{
"pp": "case basic\nα : Type u_2\nm0 : MeasurableSpace α\nμ ν : Measure α\nS T : Set (Set α)\nh_gen : m0 = generateFrom S\nhc : T.Countable\nh_inter : IsPiSystem S\nhU : ⋃₀ T = univ\nhtop : ∀ t ∈ T, μ t ≠ ∞\nST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)\nT_eq : ∀ t ∈ T, μ t = ν t\nt : Set α\nht : t ∈ T\nu✝ u... | [] | | basic u hu => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 618,
"column": 4
} | {
"line": 618,
"column": 51
} | {
"line": 619,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}\nT : Set (Set ... | [] | simpa only [← hT] using! mem_univ (default : α) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 80
} | {
"line": 75,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ (μ.trim ⋯) (f ⁻¹' s) = (Measure.map f μ) s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheo... | [] | rw [← map_trim_comap hf, Measure.map_apply (Measurable.of_comap_le le_rfl) hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 80
} | {
"line": 75,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ (μ.trim ⋯) (f ⁻¹' s) = (Measure.map f μ) s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheo... | [] | rw [← map_trim_comap hf, Measure.map_apply (Measurable.of_comap_le le_rfl) hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 80
} | {
"line": 75,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ (μ.trim ⋯) (f ⁻¹' s) = (Measure.map f μ) s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheo... | [] | rw [← map_trim_comap hf, Measure.map_apply (Measurable.of_comap_le le_rfl) hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 856,
"column": 4
} | {
"line": 856,
"column": 31
} | {
"line": 857,
"column": 2
} | [
{
"pp": "δ : Type u_4\nu : Set δ\ninst✝ : MeasureSpace δ\nhu : NullMeasurableSet u volume\n⊢ Injective Subtype.val",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype.coe_injective",
"Set.instMembership",
"Set"
],
"usedFVars": [
... | [] | exact Subtype.coe_injective | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 856,
"column": 4
} | {
"line": 856,
"column": 31
} | {
"line": 857,
"column": 2
} | [
{
"pp": "δ : Type u_4\nu : Set δ\ninst✝ : MeasureSpace δ\nhu : NullMeasurableSet u volume\n⊢ Injective Subtype.val",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype.coe_injective",
"Set.instMembership",
"Set"
],
"usedFVars": [
... | [] | exact Subtype.coe_injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 856,
"column": 4
} | {
"line": 856,
"column": 31
} | {
"line": 857,
"column": 2
} | [
{
"pp": "δ : Type u_4\nu : Set δ\ninst✝ : MeasureSpace δ\nhu : NullMeasurableSet u volume\n⊢ Injective Subtype.val",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype.coe_injective",
"Set.instMembership",
"Set"
],
"usedFVars": [
... | [] | exact Subtype.coe_injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 996,
"column": 2
} | {
"line": 997,
"column": 68
} | {
"line": 999,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf g : α → β\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\n⊢ s.piecewise f g =ᵐ[μ.restrict s] f",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
... | [] | rw [ae_restrict_eq hs]
exact (piecewise_eqOn s f g).eventuallyEq.filter_mono inf_le_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 996,
"column": 2
} | {
"line": 997,
"column": 68
} | {
"line": 999,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf g : α → β\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\n⊢ s.piecewise f g =ᵐ[μ.restrict s] f",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
... | [] | rw [ae_restrict_eq hs]
exact (piecewise_eqOn s f g).eventuallyEq.filter_mono inf_le_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 504,
"column": 6
} | {
"line": 512,
"column": 54
} | {
"line": 513,
"column": 2
} | [
{
"pp": "case refine_1.iUnion\nα : Type u_1\nt : ℕ → Set α\nn : ℕ\ns : Set α\nf : ℕ → Set α\nhs✝ : ∀ (n_1 : ℕ), MeasurableSet (f n_1)\nh : ∀ (n_1 : ℕ), ∃ S, ↑S ⊆ memPartition t n ∧ f n_1 = ⋃₀ ↑S\n⊢ ∃ S, ↑S ⊆ memPartition t n ∧ ⋃ i, f i = ⋃₀ ↑S",
"ppTerm": "?refine_1.iUnion",
"assigned": true,
"usedC... | [] | choose S hS_subset hS_eq using h
have : Fintype (⋃ n, (S n : Set (Set α))) := by
refine (Finite.subset (finite_memPartition t n) ?_).fintype
simp only [iUnion_subset_iff]
exact hS_subset
refine ⟨(⋃ n, (S n : Set (Set α))).toFinset, ?_, ?_⟩
· simp only [coe_toFinset, iUnion_subs... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 504,
"column": 6
} | {
"line": 512,
"column": 54
} | {
"line": 513,
"column": 2
} | [
{
"pp": "case refine_1.iUnion\nα : Type u_1\nt : ℕ → Set α\nn : ℕ\ns : Set α\nf : ℕ → Set α\nhs✝ : ∀ (n_1 : ℕ), MeasurableSet (f n_1)\nh : ∀ (n_1 : ℕ), ∃ S, ↑S ⊆ memPartition t n ∧ f n_1 = ⋃₀ ↑S\n⊢ ∃ S, ↑S ⊆ memPartition t n ∧ ⋃ i, f i = ⋃₀ ↑S",
"ppTerm": "?refine_1.iUnion",
"assigned": true,
"usedC... | [] | choose S hS_subset hS_eq using h
have : Fintype (⋃ n, (S n : Set (Set α))) := by
refine (Finite.subset (finite_memPartition t n) ?_).fintype
simp only [iUnion_subset_iff]
exact hS_subset
refine ⟨(⋃ n, (S n : Set (Set α))).toFinset, ?_, ?_⟩
· simp only [coe_toFinset, iUnion_subs... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Order.Lattice | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 50
} | {
"line": 214,
"column": 2
} | [
{
"pp": "α : Type u_2\nm : MeasurableSpace α\nδ : Type u_3\ninst✝² : MeasurableSpace δ\ninst✝¹ : SemilatticeSup α\ninst✝ : MeasurableSup₂ α\nf : ℕ → δ → α\nn : ℕ\nhf : ∀ k ≤ n, Measurable (f k)\n⊢ Measurable fun x ↦ (range (n + 1)).sup' ⋯ fun k ↦ f k x",
"ppTerm": "?m.28",
"assigned": true,
"usedCon... | [
"α : Type u_2\nm : MeasurableSpace α\nδ : Type u_3\ninst✝² : MeasurableSpace δ\ninst✝¹ : SemilatticeSup α\ninst✝ : MeasurableSup₂ α\nf : ℕ → δ → α\nn : ℕ\nhf : ∀ k ≤ n, Measurable (f k)\n⊢ (fun x ↦ (range (n + 1)).sup' ⋯ fun k ↦ f k x) = (range (n + 1)).sup' ⋯ f"
] | convert! Finset.measurable_range_sup' hf using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 177,
"column": 2
} | {
"line": 179,
"column": 74
} | {
"line": 181,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ : Measure α\ng : β → γ\nf : α → β\nhg : AEMeasurable g (Measure.map f μ)\nhf : AEMeasurable f μ\n⊢ Measure.map g (Measure.map f μ) = Measure.map (g ∘ f) μ",
"ppTerm": "?m.31",
... | [] | ext1 s hs
rw [map_apply_of_aemeasurable hg hs, map_apply₀ hf (hg.nullMeasurable hs),
map_apply_of_aemeasurable (hg.comp_aemeasurable hf) hs, preimage_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 177,
"column": 2
} | {
"line": 179,
"column": 74
} | {
"line": 181,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ : Measure α\ng : β → γ\nf : α → β\nhg : AEMeasurable g (Measure.map f μ)\nhf : AEMeasurable f μ\n⊢ Measure.map g (Measure.map f μ) = Measure.map (g ∘ f) μ",
"ppTerm": "?m.31",
... | [] | ext1 s hs
rw [map_apply_of_aemeasurable hg hs, map_apply₀ hf (hg.nullMeasurable hs),
map_apply_of_aemeasurable (hg.comp_aemeasurable hf) hs, preimage_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1136,
"column": 6
} | {
"line": 1136,
"column": 59
} | {
"line": 1136,
"column": 60
} | [
{
"pp": "case e_f\nα : Type u_2\nι : Type u_6\nx✝ : MeasurableSpace α\nμ : Measure α\ns : ι → Set α\nM : ℕ\nhs_meas : ∀ (i : ι), MeasurableSet (s i)\nhs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M\nt : Set α\nht : MeasurableSet t\nF : Finset ι\nP : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \\ C, (s i)ᶜ\nC... | [
"case pos\nα : Type u_2\nι : Type u_6\nx✝ : MeasurableSpace α\nμ : Measure α\ns : ι → Set α\nM : ℕ\nhs_meas : ∀ (i : ι), MeasurableSet (s i)\nhs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M\nt : Set α\nht : MeasurableSet t\nF : Finset ι\nP : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \\ C, (s i)ᶜ\nCs : Finset (... | by_cases hC : C ∈ F.powerset <;> by_cases hC' : C = ∅ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 21
} | {
"line": 66,
"column": 22
} | [
{
"pp": "case basic\nα : Type u_1\ns : Set (Set α)\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nhs : t = TopologicalSpace.generateFrom s\nu✝ u : Set α\nhu : u ∈ s\n⊢ MeasurableSet u",
"ppTerm": "?basic",
"assigned": true,
"usedConstants": [
"MeasurableSpace.GenerateMeasurable.ba... | [] | | basic u hu => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 23
} | {
"line": 66,
"column": 6
} | [
{
"pp": "α : Type u_1\ns : Set (Set α)\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nhs : t = TopologicalSpace.generateFrom s\nu : Set α\nhu : TopologicalSpace.IsOpen u\n⊢ MeasurableSet u",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
... | [] | induction hu with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 71
} | {
"line": 59,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\nthis : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)\n⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range ... | let : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 36
} | {
"line": 175,
"column": 37
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\nE : Set α\n⊢ 0 < infEDist x E ↔ x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreor... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\nE : Set α\n⊢ 0 < infEDist x E ↔ ¬infEDist x E = 0"
] | mem_closure_iff_infEDist_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 56
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nx : α\nδ : ℝ\nhδ : δ < 0\n⊢ closedBall x δ ⊆ cthickening δ {x}",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"Set.instSingletonS... | [] | simp only [closedBall_eq_empty.mpr hδ, empty_subset] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 56
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nx : α\nδ : ℝ\nhδ : δ < 0\n⊢ closedBall x δ ⊆ cthickening δ {x}",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"Set.instSingletonS... | [] | simp only [closedBall_eq_empty.mpr hδ, empty_subset] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 56
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nx : α\nδ : ℝ\nhδ : δ < 0\n⊢ closedBall x δ ⊆ cthickening δ {x}",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"Set.instSingletonS... | [] | simp only [closedBall_eq_empty.mpr hδ, empty_subset] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 60
} | {
"line": 578,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\n⊢ infDist x s = ⨅ y, dist x ↑y",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real",
"iInf",
"congrArg",
"CompletelyDistribLatt... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\n⊢ ⨅ i, (edist x ↑i).toReal = ⨅ y, dist x ↑y",
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\n⊢ ∀ (i : Subtype (Membership.mem s)), edist x ↑i ≠ ∞"
] | rw [infDist, infEDist, iInf_subtype', ENNReal.toReal_iInf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 878,
"column": 2
} | {
"line": 878,
"column": 19
} | {
"line": 880,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\nx : α\nr : ℝ\nh : x ∈ s\nH : hausdorffDist s t < r\nfin : hausdorffEDist s t ≠ ∞\nr0 : 0 < r\nthis : hausdorffEDist s t < ENNReal.ofReal r\ny : α\nhy : y ∈ t\nyr : dist x y < r\n⊢ ∃ y ∈ t, dist x y < r",
"ppTerm": "?m.114",
"assigned": true,... | [] | exact ⟨y, hy, yr⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.Prod | {
"line": 42,
"column": 6
} | {
"line": 47,
"column": 40
} | {
"line": 48,
"column": 2
} | [
{
"pp": "case a.refine_2\nα : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set β\nhs : s ∈ D\n⊢ MeasurableSet (Prod.snd ⁻¹' s)",
"ppTerm": "?a.refine_2✝",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mp... | [] | rcases hC with ⟨t, h1t, h2t⟩
rw [← univ_prod, ← h2t, iUnion_prod_const]
apply MeasurableSet.iUnion
rintro n
apply measurableSet_generateFrom
exact mem_image2_of_mem (h1t n) hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Prod | {
"line": 42,
"column": 6
} | {
"line": 47,
"column": 40
} | {
"line": 48,
"column": 2
} | [
{
"pp": "case a.refine_2\nα : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set β\nhs : s ∈ D\n⊢ MeasurableSet (Prod.snd ⁻¹' s)",
"ppTerm": "?a.refine_2✝",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mp... | [] | rcases hC with ⟨t, h1t, h2t⟩
rw [← univ_prod, ← h2t, iUnion_prod_const]
apply MeasurableSet.iUnion
rintro n
apply measurableSet_generateFrom
exact mem_image2_of_mem (h1t n) hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 74,
"column": 2
} | {
"line": 78,
"column": 75
} | {
"line": 80,
"column": 0
} | [
{
"pp": "⊢ generateFrom (range fun a ↦ Ioi ↑a) = generateFrom (range fun a ↦ Iic ↑a)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.mem_range_self",
"Eq.mpr",
"Real",
"Set.Ioi",
"Lattice.toSemilatticeSup",
"MeasurableSet",
"congrArg",
"... | [] | refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>
rintro _ ⟨q, rfl⟩ <;>
dsimp only <;>
[rw [← compl_Iic]; rw [← compl_Ioi]] <;>
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q)) | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 213,
"column": 40
} | {
"line": 213,
"column": 68
} | {
"line": 214,
"column": 6
} | [
{
"pp": "case hunion\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Co... | [
"case hunion\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Continuous[ins... | ⟨g', s', s'_meas, t's', hg'⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 42
} | {
"line": 256,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ : SigmaFinite (μ.restrict t)\n⊢ FinStronglyMeasurable f μ",
"ppTerm": "?m.26",... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ this : SigmaFinite (μ.restrict t)\n⊢ FinStronglyMeasurable f μ"
] | have : SigmaFinite (μ.restrict t) := htμ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 262,
"column": 8
} | {
"line": 262,
"column": 58
} | {
"line": 262,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ this : SigmaFinite (μ.restrict t)\nS : ℕ → Set α := spanningSets (μ.restrict t)\nh... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ this : SigmaFinite (μ.restrict t)\nS : ℕ → Set α := spanningSets (μ.restrict t)\nhS_meas : ∀ (... | SimpleFunc.restrict_apply _ ((hS_meas n).inter ht) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 31
} | {
"line": 78,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nhf : ∀ a ∈ s, a ∈ t → f a ≤ 1\nhf' : ∀ a ∈ s, a ∉ t → f a = 0\n⊢ ∫⁻ (a : α) in s, f a ∂μ ≤ μ t",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nhf : ∀ a ∈ s, a ∈ t → f a ≤ 1\nhf' : ∀ a ∈ s, a ∉ t → f a = 0\n⊢ ∫⁻ (a : α), s.indicator f a ∂μ ≤ μ t"
] | rw [← lintegral_indicator hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 455,
"column": 24
} | {
"line": 457,
"column": 35
} | {
"line": 459,
"column": 0
} | [
{
"pp": "α : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : IsEmpty α\nμ : Measure α\nf : α → ℝ≥0∞\n⊢ ∫⁻ (x : α), f x ∂μ = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HEq.refl",
"inferInstance",
"Lean.Meta.FastSubsingle... | [] | by
have : Subsingleton (Measure α) := inferInstance
convert! lintegral_zero_measure f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 467,
"column": 2
} | {
"line": 468,
"column": 38
} | {
"line": 471,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhs' : μ s = 0\n⊢ ∫⁻ (x : α) in s, f x ∂μ = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MeasureTheory.Measure",
"HEq.refl",
"MeasureTheory.Measure.... | [] | convert! lintegral_zero_measure _
exact Measure.restrict_eq_zero.2 hs' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 467,
"column": 2
} | {
"line": 468,
"column": 38
} | {
"line": 471,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhs' : μ s = 0\n⊢ ∫⁻ (x : α) in s, f x ∂μ = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MeasureTheory.Measure",
"HEq.refl",
"MeasureTheory.Measure.... | [] | convert! lintegral_zero_measure _
exact Measure.restrict_eq_zero.2 hs' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 488,
"column": 2
} | {
"line": 488,
"column": 20
} | {
"line": 489,
"column": 2
} | [
{
"pp": "case e_f\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\ng : α →ₛ ℝ≥0∞\nhg : ⇑g ≤ fun a ↦ s.indicator f a\nthis : ⇑g ≤ f\nt : ℝ≥0∞\n⊢ t * μ (⇑g ⁻¹' {t}) = t * μ (⇑g ⁻¹' {t} ∩ s)",
"ppTerm": "?e_f",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measu... | [
"case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\ng : α →ₛ ℝ≥0∞\nhg : ⇑g ≤ fun a ↦ s.indicator f a\nthis : ⇑g ≤ f\nt : ℝ≥0∞\nH : t = 0\n⊢ t * μ (⇑g ⁻¹' {t}) = t * μ (⇑g ⁻¹' {t} ∩ s)",
"case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\ng : ... | by_cases H : t = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 671,
"column": 2
} | {
"line": 671,
"column": 27
} | {
"line": 672,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nhf : StronglyMeasurable f\nthis : IsSeparable (closure[inst✝] (⋃ n, range ⇑(hf.approx n)))\nx : α\nn : ℕ\n⊢ (hf.approx n) x ∈ ⋃ n, range ⇑(hf.approx n)",
"ppTerm": "?m.89",
"assigned": true,
"usedConst... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nhf : StronglyMeasurable f\nthis : IsSeparable (closure[inst✝] (⋃ n, range ⇑(hf.approx n)))\nx : α\nn : ℕ\n⊢ (hf.approx n) x ∈ range ⇑(hf.approx n)"
] | apply mem_iUnion_of_mem n | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 308,
"column": 10
} | {
"line": 308,
"column": 49
} | {
"line": 308,
"column": 50
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (⨆ n, (eapprox f n).lintegral μ) + ⨆ n, (eapprox g n).lintegral μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"MeasureTh... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (⨆ n, (eapprox f n).lintegral μ) + ⨆ n, (eapprox g n).lintegral μ =\n (⨆ n, (eapprox f n).lintegral μ) + ∫⁻ (a : α), g a ∂μ"
] | lintegral_eq_iSup_eapprox_lintegral hf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 29
} | {
"line": 81,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nh : ¬∃ s, MeasurableSet s ∧ SigmaFinite (μ.restrict s) ∧ ∀ t ⊆ sᶜ, ν t ≠ 0 → μ t = ∞\n⊢ MeasurableSet ∅",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"MeasurableSet.empty"
],
"usedFVars": [
"α",... | [] | · exact MeasurableSet.empty | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 780,
"column": 4
} | {
"line": 781,
"column": 57
} | {
"line": 783,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : PseudoMetrizableSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : PseudoMetrizableSpace γ\ng : β → γ\nf : α → β\nhg : IsEmbedding g\nthis : PseudoMetricSpace γ := pseudoMetrizableSpacePseud... | [] | have : IsSeparable (g ⁻¹' range (g ∘ f)) := hg.isSeparable_preimage H.isSeparable_range
rwa [range_comp, hg.injective.preimage_image] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 780,
"column": 4
} | {
"line": 781,
"column": 57
} | {
"line": 783,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : PseudoMetrizableSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : PseudoMetrizableSpace γ\ng : β → γ\nf : α → β\nhg : IsEmbedding g\nthis : PseudoMetricSpace γ := pseudoMetrizableSpacePseud... | [] | have : IsSeparable (g ⁻¹' range (g ∘ f)) := hg.isSeparable_preimage H.isSeparable_range
rwa [range_comp, hg.injective.preimage_image] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 799,
"column": 4
} | {
"line": 799,
"column": 29
} | {
"line": 800,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nι : Type u_5\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : PseudoMetrizableSpace β\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → β\ng : α → β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nlim : ∀ (x : α), Tendsto (fu... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\nι : Type u_5\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : PseudoMetrizableSpace β\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → β\ng : α → β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nlim : ∀ (x : α), Tendsto (fun i ↦ f i x)... | apply mem_iUnion_of_mem n | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 131,
"column": 60
} | {
"line": 131,
"column": 62
} | {
"line": 131,
"column": 63
} | [
{
"pp": "α : 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_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 ... | [
"α : 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_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, ... | ha | Lean.Elab.Tactic.evalIntro | ident |
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