module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Logic.Hydra | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 41
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : Std.Irrefl r\ns' s : Multiset α\n⊢ (∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ s' = (s + t).erase a) ↔\n ∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t",
"usedConstants": [
"Membership.mem",
"Multiset",
"Multi... | refine exists₂_congr fun t a ↦ ⟨?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Logic.Hydra | {
"line": 161,
"column": 13
} | {
"line": 161,
"column": 70
} | [
{
"pp": "case empty\nα : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\nhs : ∀ a ∈ 0, Acc (CutExpand r) {a}\n⊢ Acc (CutExpand r) 0",
"usedConstants": [
"False.elim",
"Relation.CutExpand",
"Multiset",
"Acc",
"Zero.toOfNat0",
"OfNat.ofNat",
"Relation.not_cutExpand_... | exact Acc.intro 0 fun s h ↦ (not_cutExpand_zero s h).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Logic.Hydra | {
"line": 161,
"column": 13
} | {
"line": 161,
"column": 70
} | [
{
"pp": "case empty\nα : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\nhs : ∀ a ∈ 0, Acc (CutExpand r) {a}\n⊢ Acc (CutExpand r) 0",
"usedConstants": [
"False.elim",
"Relation.CutExpand",
"Multiset",
"Acc",
"Zero.toOfNat0",
"OfNat.ofNat",
"Relation.not_cutExpand_... | exact Acc.intro 0 fun s h ↦ (not_cutExpand_zero s h).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Hydra | {
"line": 161,
"column": 13
} | {
"line": 161,
"column": 70
} | [
{
"pp": "case empty\nα : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\nhs : ∀ a ∈ 0, Acc (CutExpand r) {a}\n⊢ Acc (CutExpand r) 0",
"usedConstants": [
"False.elim",
"Relation.CutExpand",
"Multiset",
"Acc",
"Zero.toOfNat0",
"OfNat.ofNat",
"Relation.not_cutExpand_... | exact Acc.intro 0 fun s h ↦ (not_cutExpand_zero s h).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 361,
"column": 29
} | {
"line": 361,
"column": 51
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : Field K\ninst✝¹² : CharZero K\ninst✝¹¹ : DecidableEq ι\ninst✝¹⁰ : Fintype ι\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module K M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module K N\nP : RootPairing ι K M N\ninst✝⁵ : P.IsRootSystem\ninst✝⁴ : P.IsCryst... | obtain ⟨i, hi⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 428,
"column": 8
} | {
"line": 428,
"column": 74
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Module.Free R V\ninst✝ : Module.Finite R V\ng : ↥(Subgroup.center (SpecialLinearGroup R V))\nhR : Nontrivial R\nhV : 1 ≤ Module.finrank R V\nr : R := ⋯.choose\nhr : r ^ max (Module.finrank R V) 1 = 1... | exact ⟨this.unit, by simp [mem_rootsOfUnity, ← Units.val_inj, hr]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 144,
"column": 4
} | {
"line": 147,
"column": 22
} | [
{
"pp": "case a\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → MeasurableSpace (α i)\ni : ι\n⊢ ((fun t ↦ (↑{i}).pi t) '' univ.pi fun i ↦ {s | MeasurableSet s}) ⊆\n ⋃ s, (fun t ↦ (↑s).pi t) '' univ.pi fun i ↦ {s | MeasurableSet s}",
"usedConstants": [
"MeasurableSet",
"Finset",
"Set.... | exact subset_iUnion
(fun (s : Finset ι) ↦
(fun t : ∀ i, Set (α i) ↦ (s : Set ι).pi t) '' univ.pi (fun i ↦ setOf MeasurableSet))
({i} : Finset ι) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 209,
"column": 41
} | {
"line": 210,
"column": 49
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ns : Finset ι\nS : Set ((i : ↥s) → α ↑i)\n⊢ (cylinder s S)ᶜ = cylinder s Sᶜ",
"usedConstants": [
"Set.ext",
"_private.Mathlib.MeasureTheory.Constructions.Cylinders.0.MeasureTheory.compl_cylinder._simp_1_1",
"congrArg",
"Compl.compl",
"Fin... | by
ext1 f; simp only [mem_compl_iff, mem_cylinder] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 21
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ns : Finset ι\nS : Set ((i : ↥s) → α ↑i)\nhS : MeasurableSet S\n⊢ (cylinder s S)ᶜ = cylinder s Sᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Finset",
"MeasureTheory.compl_cylinder",... | rw [compl_cylinder] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order.WithTop | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 53
} | [
{
"pp": "case inr.a.inr.coe.coe\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι... | simp only [mem_Iio, WithTop.coe_lt_coe] at hb | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order.WithTop | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 76
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nα : Type u_2\nf : Filter α\nx : α → WithTop ι\nh : Nonempty ι\n⊢ (∀ i < ⊤, ∀ᶠ (x_1 : α) in f, x x_1 ∈ Ioi i) ↔ ∀ (i : ι), ∀ᶠ (a : α) in f, ↑i < x a",
"usedConstants": [
"Eq.mpr",
"Set.I... | rw [← Set.forall_mem_range (p := (∀ᶠ a in f, · < x a)), WithTop.range_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 205,
"column": 80
} | {
"line": 205,
"column": 97
} | [
{
"pp": "α : Type u_1\n𝒜 : Set (Set α)\ns u : Set α\nhs✝ : generateSetAlgebra 𝒜 u\nA : Set (Set (Set α))\nA_fin : A.Finite\nmem_A : ∀ a ∈ A, a.Finite\nhA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜\nu_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t\nthis✝ : Finite ↑A\nthis : ∀ (a : ↑A), Finite ↑↑a\nx : α\nf : (i : Set (Set α)) → i ∈ A... | exact (hf a ha).2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 96,
"column": 69
} | {
"line": 98,
"column": 31
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\n⊢ μ.toSphere = 0 ↔ dim E = 0",
"usedConstants": [
"Eq.mpr",
"False",
"Real.partialOr... | by
rw [← measure_univ_eq_zero, toSphere_apply_univ]
simp [IsOpen.measure_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 36
} | [
{
"pp": "n : ℕ\nx : ↑(Ioi 0)\n⊢ (volumeIoiPow n) (Iio x) = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))",
"usedConstants": [
"Real.instLE",
"Real",
"Set.Ioi",
"Real.instZero",
"le_of_lt",
"Membership.mem",
"LE.le",
"Zero.toOfNat0",
"OfNat.ofNat",
"Subt... | have hr₀ : 0 ≤ x.1 := le_of_lt x.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 15
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\na✝ : Nontrivial E\ns : Set ↑(sphere 0 1)\nhs : s ∈ {s | MeasurableSet s}\nr : ↑(Ioi 0)\nthis : Ioo 0 ↑r = ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 247,
"column": 2
} | {
"line": 248,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nε : ℝ\nhε : 0 < ε\nx : ↑(sphere 0 1)\n⊢ ↑(toSphereBallBound (dim E) ε) * μ.real (ball 0 1) ≤ μ.toSphere.re... | grw [Measure.real, Measure.real, ← toSphereBallBound_mul_measure_unitBall_le_toSphere_ball μ hε,
ENNReal.toReal_mul, ENNReal.coe_toReal] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 247,
"column": 2
} | {
"line": 249,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nε : ℝ\nhε : 0 < ε\nx : ↑(sphere 0 1)\n⊢ ↑(toSphereBallBound (dim E) ε) * μ.real (ball 0 1) ≤ μ.toSphere.re... | grw [Measure.real, Measure.real, ← toSphereBallBound_mul_measure_unitBall_le_toSphere_ball μ hε,
ENNReal.toReal_mul, ENNReal.coe_toReal]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 247,
"column": 2
} | {
"line": 249,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nε : ℝ\nhε : 0 < ε\nx : ↑(sphere 0 1)\n⊢ ↑(toSphereBallBound (dim E) ε) * μ.real (ball 0 1) ≤ μ.toSphere.re... | grw [Measure.real, Measure.real, ← toSphereBallBound_mul_measure_unitBall_le_toSphere_ball μ hε,
ENNReal.toReal_mul, ENNReal.coe_toReal]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.SetSemiring | {
"line": 443,
"column": 8
} | {
"line": 443,
"column": 36
} | [
{
"pp": "case h.refine_1.h\nα : Type u_1\nC : Set (Set α)\nJ✝ : Finset (Set α)\nhC : IsSetSemiring C\ns : Set α\nJ : Finset (Set α)\nhJ : s ∉ J\nhind :\n ↑J ⊆ C →\n ∃ K,\n (↑J).PairwiseDisjoint K ∧\n (∀ i ∈ J, ↑(K i) ⊆ C) ∧\n (⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧\n (∀ j ∈ J... | exact ⟨h8, Disjoint.symm h8⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 288,
"column": 24
} | {
"line": 288,
"column": 35
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : Nontrivial E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nf : ℝ → F\n⊢... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 100,
"column": 35
} | {
"line": 100,
"column": 56
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed (s i)\nr₁ r₂ : ℕ → ℝ\n... | (hs (f j)).closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 197,
"column": 6
} | {
"line": 198,
"column": 17
} | [
{
"pp": "case h_mem\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetSemiring C\nJ J' : Finset (Set α)\nhJ : ↑J ⊆ C\nhJdisj : (↑J).PairwiseDisjoint id\nhJ' : ↑J' ⊆ C\nhJ'disj : (↑J').PairwiseDisjoint id\nh : ⋃₀ ↑J = ⋃₀ ↑J'\ns : Set α\nhs : s ∈ J\nthis : s = ⋃ ... | rw [← this]
exact hJ hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 197,
"column": 6
} | {
"line": 198,
"column": 17
} | [
{
"pp": "case h_mem\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetSemiring C\nJ J' : Finset (Set α)\nhJ : ↑J ⊆ C\nhJdisj : (↑J).PairwiseDisjoint id\nhJ' : ↑J' ⊆ C\nhJ'disj : (↑J').PairwiseDisjoint id\nh : ⋃₀ ↑J = ⋃₀ ↑J'\ns : Set α\nhs : s ∈ J\nthis : s = ⋃ ... | rw [← this]
exact hJ hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 248,
"column": 8
} | {
"line": 249,
"column": 43
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns✝ t : Set α\nI✝ : Finset (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm✝ m' m : AddContent G C\nhC : IsSetSemiring C\nI : Finset (Set α)\nhI : ↑I ⊆ _root_.supClosure C\nh'I : (↑I).PairwiseDisjoint id\nhh'I : ⋃₀ ↑I ∈ _root_.supClosure C\ns : Set α\nhs : s ∈ I\n⊢ ∃ P, ↑... | have := hI hs
rwa [hC.mem_supClosure_iff] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 248,
"column": 8
} | {
"line": 249,
"column": 43
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns✝ t : Set α\nI✝ : Finset (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm✝ m' m : AddContent G C\nhC : IsSetSemiring C\nI : Finset (Set α)\nhI : ↑I ⊆ _root_.supClosure C\nh'I : (↑I).PairwiseDisjoint id\nhh'I : ⋃₀ ↑I ∈ _root_.supClosure C\ns : Set α\nhs : s ∈ I\n⊢ ∃ P, ↑... | have := hI hs
rwa [hC.mem_supClosure_iff] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.SetSemiring | {
"line": 534,
"column": 6
} | {
"line": 536,
"column": 55
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Nonempty α\nu v : α\nhuv : u ≤ v\nu' v' : α\nhu'v' : u' ≤ v'\nhu : u' ≤ u\n⊢ ∃ I, ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} ∧ (↑I).PairwiseDisjoint id ∧ Set.Ioc u v \\ Set.Ioc u' v' = ⋃₀ ↑I",
"usedConstants": [
"Set.Ioc",
"Lattice.... | rcases Ioc_mem_setOf_Ioc_le (max u v') v with ⟨u'', v'', h'', heq⟩
exists {Set.Ioc u'' v''}
grind [coe_singleton, pairwiseDisjoint_singleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.SetSemiring | {
"line": 534,
"column": 6
} | {
"line": 536,
"column": 55
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Nonempty α\nu v : α\nhuv : u ≤ v\nu' v' : α\nhu'v' : u' ≤ v'\nhu : u' ≤ u\n⊢ ∃ I, ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} ∧ (↑I).PairwiseDisjoint id ∧ Set.Ioc u v \\ Set.Ioc u' v' = ⋃₀ ↑I",
"usedConstants": [
"Set.Ioc",
"Lattice.... | rcases Ioc_mem_setOf_Ioc_le (max u v') v with ⟨u'', v'', h'', heq⟩
exists {Set.Ioc u'' v''}
grind [coe_singleton, pairwiseDisjoint_singleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 43,
"column": 8
} | {
"line": 43,
"column": 45
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : f =ᶠ[ae (μ.restrict s)] 0\nhm : ¬m ≤ m0\n⊢ μ[f | m] =ᶠ[ae (μ.restrict s)] 0",
"usedConsta... | · simp_rw [condExp_of_not_le hm]; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 476,
"column": 8
} | {
"line": 480,
"column": 36
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG✝ : Type u_2\ninst✝² : AddCommMonoid G✝\nm m' : AddContent G✝ C\ninst✝¹ : LinearOrder α\nG : Type u_3\ninst✝ : AddCommGroup G\nf : α → G\nn : ℕ\nih :\n ∀ (I : Finset (Set α)),\n ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →\n (↑I)... | have : (Ioc u' v ∪ ⋃₀ ↑I') \ Ioc u' v = ⋃₀ ↑I' := by
refine Disjoint.sup_sdiff_cancel_left ?_
simp only [coe_erase, disjoint_sUnion_right, mem_diff, mem_singleton_iff, and_imp, I']
intro u hu hu'
exact (h'I hu tI hu').symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 97,
"column": 40
} | {
"line": 97,
"column": 71
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ... | filter_upwards [lem1] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 197,
"column": 51
} | {
"line": 197,
"column": 65
} | [
{
"pp": "F : Type u_2\ninst✝ : SeminormedAddCommGroup F\na b : ℝ\nf : ℝ → F\nhf : AbsolutelyContinuousOnInterval f a b\n⊢ Tendsto ?m.10 (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)",
"usedConstants": []
}
] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 197,
"column": 51
} | {
"line": 197,
"column": 65
} | [
{
"pp": "F : Type u_2\ninst✝ : SeminormedAddCommGroup F\na b : ℝ\nf : ℝ → F\nhf : AbsolutelyContinuousOnInterval f a b\n⊢ Tendsto ?m.10 (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)",
"usedConstants": []
}
] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 197,
"column": 51
} | {
"line": 197,
"column": 65
} | [
{
"pp": "F : Type u_2\ninst✝ : SeminormedAddCommGroup F\na b : ℝ\nf : ℝ → F\nhf : AbsolutelyContinuousOnInterval f a b\n⊢ Tendsto ?m.10 (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)",
"usedConstants": []
}
] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 644,
"column": 2
} | {
"line": 646,
"column": 79
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nhC : IsSetRing C\nm : AddContent ℝ≥0∞ C\nhm_ne_top : ∀ s ∈ C, m s ≠ ∞\nhm_tendsto : ∀ ⦃s : ℕ → Set α⦄, (∀ (n : ℕ), s n ∈ C) → Antitone s → ⋂ n, s n = ∅ → Tendsto (fun n ↦ m (s n)) atTop (𝓝 0)\nf : ℕ → Set α\nhf : ∀ (i : ℕ), f i ∈ C\nhUf : ⋃ i, f i ∈ C\nh_disj : Pairwise ... | have hmsn n : m (s n) = m (⋃ i, f i) - ∑ i ∈ Finset.range (n + 1), m (f i) := by
rw [addContent_diff_of_ne_top m hC hm_ne_top hUf (hC.accumulate_mem hf n)
(Set.accumulate_subset_iUnion _), addContent_accumulate m hC h_disj hf n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 182,
"column": 4
} | {
"line": 186,
"column": 31
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nS : Set α\nhS₁ : MeasurableSet S\nhS₂ : j.posPart S = 0\nhS₃ : j.negPart Sᶜ = 0\n⊢ 0 ≤[Sᶜ] j.toSignedMeasure",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.JordanDecomposition.po... | refine restrict_le_restrict_of_subset_le _ _ fun A hA hA₁ => ?_
rw [toSignedMeasure, toSignedMeasure_sub_apply hA, measureReal_def (μ := j.negPart),
show j.negPart A = 0 from nonpos_iff_eq_zero.1 (hS₃ ▸ measure_mono hA₁), ENNReal.toReal_zero,
sub_zero]
exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 182,
"column": 4
} | {
"line": 186,
"column": 31
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nS : Set α\nhS₁ : MeasurableSet S\nhS₂ : j.posPart S = 0\nhS₃ : j.negPart Sᶜ = 0\n⊢ 0 ≤[Sᶜ] j.toSignedMeasure",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.JordanDecomposition.po... | refine restrict_le_restrict_of_subset_le _ _ fun A hA hA₁ => ?_
rw [toSignedMeasure, toSignedMeasure_sub_apply hA, measureReal_def (μ := j.negPart),
show j.negPart A = 0 from nonpos_iff_eq_zero.1 (hS₃ ▸ measure_mono hA₁), ENNReal.toReal_zero,
sub_zero]
exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal... | refine le_trans (eLpNorm_add_le ?_ ?_ hp_one) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 112,
"column": 2
} | {
"line": 119,
"column": 35
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\nv w : VectorMeasure α M\n⊢ v = w ↔ ∀ (i : Set α), MeasurableSet i → ↑v i = ↑w i",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"AddMonoid.toAddZeroClass",
... | constructor
· rintro rfl _ _
rfl
· rw [ext_iff']
intro h i
by_cases hi : MeasurableSet i
· exact h i hi
· simp_rw [not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 112,
"column": 2
} | {
"line": 119,
"column": 35
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\nv w : VectorMeasure α M\n⊢ v = w ↔ ∀ (i : Set α), MeasurableSet i → ↑v i = ↑w i",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"AddMonoid.toAddZeroClass",
... | constructor
· rintro rfl _ _
rfl
· rw [ext_iff']
intro h i
by_cases hi : MeasurableSet i
· exact h i hi
· simp_rw [not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 207,
"column": 2
} | {
"line": 218,
"column": 53
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\nι : Type u_4\ns : Finset ι\nf : ι → Set α\nhd : (↑s).PairwiseDisjoint f\nhm : ∀ b ∈ s, MeasurableSet (f b)\n⊢ ↑v (⋃ b ∈ s, f b) = ∑ p ∈ s, ↑v (f p)",
"... | induction s using Finset.induction with
| empty => simp
| insert a s has ih =>
simp only [Finset.mem_insert, iUnion_iUnion_eq_or_left, has, not_false_eq_true,
Finset.sum_insert]
rw [of_union, ih]
· exact hd.subset (by simp)
· grind
· simp only [disjoint_iUnion_right]
exact fun i hi ↦... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 207,
"column": 2
} | {
"line": 218,
"column": 53
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\nι : Type u_4\ns : Finset ι\nf : ι → Set α\nhd : (↑s).PairwiseDisjoint f\nhm : ∀ b ∈ s, MeasurableSet (f b)\n⊢ ↑v (⋃ b ∈ s, f b) = ∑ p ∈ s, ↑v (f p)",
"... | induction s using Finset.induction with
| empty => simp
| insert a s has ih =>
simp only [Finset.mem_insert, iUnion_iUnion_eq_or_left, has, not_false_eq_true,
Finset.sum_insert]
rw [of_union, ih]
· exact hd.subset (by simp)
· grind
· simp only [disjoint_iUnion_right]
exact fun i hi ↦... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 207,
"column": 2
} | {
"line": 218,
"column": 53
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\nι : Type u_4\ns : Finset ι\nf : ι → Set α\nhd : (↑s).PairwiseDisjoint f\nhm : ∀ b ∈ s, MeasurableSet (f b)\n⊢ ↑v (⋃ b ∈ s, f b) = ∑ p ∈ s, ↑v (f p)",
"... | induction s using Finset.induction with
| empty => simp
| insert a s has ih =>
simp only [Finset.mem_insert, iUnion_iUnion_eq_or_left, has, not_false_eq_true,
Finset.sum_insert]
rw [of_union, ih]
· exact hd.subset (by simp)
· grind
· simp only [disjoint_iUnion_right]
exact fun i hi ↦... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nh : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x\nf : ℕ → ℝ\nhf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n\nn : ℕ\nhlt : f n < -↑n\nB : Set α\nhB₂ : ↑s B = f n\nhB₁ : MeasurableSet B\nhBr : s ≤[B] 0\n⊢ ∃ B, MeasurableSet B ∧ s ≤... | exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 565,
"column": 2
} | {
"line": 565,
"column": 22
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\nv : VectorMeasure α M\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ ↑(v.map f) s = ↑v (f ⁻¹' s)",
"usedConstants": [
"Eq.... | rw [map, dif_pos hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 73,
"column": 4
} | {
"line": 74,
"column": 35
} | [
{
"pp": "case pos.h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → E\nhf : Integrable f μ\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(μ.withDensityᵥ (-f)) i = ↑(-μ.withDensityᵥ f) i",
"usedConstants": [
"Eq.mpr",
"NegZ... | rw [VectorMeasure.neg_apply, withDensityᵥ_apply hf hi, ← integral_neg,
withDensityᵥ_apply hf.neg hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 859,
"column": 2
} | {
"line": 859,
"column": 47
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv w : VectorMeasure α M\ni : Set α\nh : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑v j ≤ ↑w j\nhi : MeasurableSet i\n⊢ v ≤[i] w",
"usedConstants": [
"Iff.... | · exact (restrict_le_restrict_iff _ _ hi).2 h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 923,
"column": 2
} | {
"line": 923,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid M\ninst✝³ : PartialOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : OrderClosedTopology M\nv w : VectorMeasure α M\ninst✝ : Countable β\nf : β → Set α\nhf₁ : ∀ (b : β), MeasurableSet (f ... | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 58
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : IsFiniteMeasure μ\ng : α → ℝ\nhint : Integrable g μ\nℱ : ι → MeasurableSpace α\nhℱ : ∀ (i : ι), ℱ i ≤ m0\nA : MeasurableSpace α := m0\nhmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖μ[g | ℱ n] x‖₊}\n⊢ UniformIntegrable (fu... | have hg : MemLp g 1 μ := memLp_one_iff_integrable.2 hint | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 91
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : IsFiniteMeasure μ\ng : α → ℝ\nhint : Integrable g μ\nℱ : ι → MeasurableSpace α\nhℱ : ∀ (i : ι), ℱ i ≤ m0\nA : MeasurableSpace α := m0\nhmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖μ[g | ℱ n] x‖₊}\nhg : MemLp g 1 μ\n⊢ Uni... | refine uniformIntegrable_of le_rfl ENNReal.one_ne_top
(fun n => (stronglyMeasurable_condExp.mono (hℱ n)).aestronglyMeasurable) fun ε hε => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 74
} | [
{
"pp": "case refine_1\nΩ : Type u_1\ninst✝⁴ : TopologicalSpace Ω\ninst✝³ : MeasurableSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_2\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ... | · simpa only [Function.comp_apply, ENNReal.coe_le_coe] using fs_le_const | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 792,
"column": 8
} | {
"line": 793,
"column": 22
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x... | rw [Set.indicator_of_mem, zero_add]
simpa using hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 792,
"column": 8
} | {
"line": 793,
"column": 22
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x... | rw [Set.indicator_of_mem, zero_add]
simpa using hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1332,
"column": 2
} | {
"line": 1332,
"column": 35
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\n⊢ μ.toSignedMeasure.toMeasureOfZeroLE univ ⋯ ⋯ = μ",
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real",
"MeasurableSet",
"Set.univ",
"PartialOrder.toPreorder",
"PseudoMet... | refine Measure.ext fun i hi => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 74
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : HasOuterApproxClosed Ω\ninst✝¹ : BorelSpace Ω\nμ ν : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν\nkey : ∀ {F : Set Ω}, IsClosed F → μ F = ν F\n⊢ inst✝⁴ = Measura... | rw [BorelSpace.measurable_eq (α := Ω), borel_eq_generateFrom_isClosed] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 74
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : HasOuterApproxClosed Ω\ninst✝¹ : BorelSpace Ω\nμ ν : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν\nkey : ∀ {F : Set Ω}, IsClosed F → μ F = ν F\n⊢ inst✝⁴ = Measura... | rw [BorelSpace.measurable_eq (α := Ω), borel_eq_generateFrom_isClosed] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 74
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : HasOuterApproxClosed Ω\ninst✝¹ : BorelSpace Ω\nμ ν : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν\nkey : ∀ {F : Set Ω}, IsClosed F → μ F = ν F\n⊢ inst✝⁴ = Measura... | rw [BorelSpace.measurable_eq (α := Ω), borel_eq_generateFrom_isClosed] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 295,
"column": 2
} | {
"line": 298,
"column": 80
} | [
{
"pp": "case inr.h\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : HasOuterApproxClosed Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nhne : L.NeBot\nε... | have room₂ :
(lintegral (μ : Measure Ω) fun a ↦ fs M a) <
(lintegral (μ : Measure Ω) fun a ↦ fs M a) + ε / 2 :=
ENNReal.lt_add_right (ne_of_lt ((fs M).lintegral_lt_top_of_nnreal _)) ε_pos' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 888,
"column": 2
} | {
"line": 888,
"column": 11
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\ninst✝⁶ : MeasurableSpace Ω\ninst✝⁵ : MeasurableSpace Ω'\ninst✝⁴ : TopologicalSpace Ω\ninst✝³ : TopologicalSpace Ω'\ninst✝² : BorelSpace Ω\ninst✝¹ : BorelSpace Ω'\ninst✝ : NormalSpace Ω'\nf : Ω → Ω'\nhf : IsClosedEmbedding f\nμ : FiniteMeasure Ω'\nhμ : μ ∈ {μ | μ (range f)ᶜ ... | rw [B hμ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 705,
"column": 4
} | {
"line": 705,
"column": 56
} | [
{
"pp": "γ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f x) (f y) ≤ C) →... | exact isCoboundedUnder_le_of_le F (x := 0) (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 705,
"column": 4
} | {
"line": 705,
"column": 56
} | [
{
"pp": "γ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f x) (f y) ≤ C) →... | exact isCoboundedUnder_le_of_le F (x := 0) (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 705,
"column": 4
} | {
"line": 705,
"column": 56
} | [
{
"pp": "γ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f x) (f y) ≤ C) →... | exact isCoboundedUnder_le_of_le F (x := 0) (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 570,
"column": 21
} | {
"line": 570,
"column": 24
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s κ μ\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ x ∈ p1, f1 x ∈ s x\... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 176,
"column": 2
} | {
"line": 180,
"column": 13
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nmβ : (i : ι) → MeasurableSpace (β i)\nf g : (i : ι) → Ω → β i\nhf :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i))... | have A i (hi : i ∈ S) : (κ a) (g i ⁻¹' sets i) = (κ a) (f i ⁻¹' sets i) := by
apply measure_congr
filter_upwards [ha i hi] with ω hω
change (g i ω ∈ sets i) = (f i ω ∈ sets i)
simp [hω] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 31
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nmβ : (i : ι) → MeasurableSpace (β i)\nf g : (i : ι) → Ω → β i\nh : ∀ (i : ι), ∀ᵐ (a : α) ∂μ, f i =ᶠ[ae (κ a)] g i\nh' : iIndepFun g κ μ\n⊢ iIndepFun f κ μ",
"us... | refine h'.congr' fun i ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 637,
"column": 4
} | {
"line": 637,
"column": 40
} | [
{
"pp": "α : Type u_1\nι : Type u_3\n_mα : MeasurableSpace α\nΩ : Type u_4\nm : ι → MeasurableSpace Ω\nm' m0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\nh_le : ∀ (i : ι), m i ≤ m0\nh_le' : m' ≤ m0\nhm : Directed (fun x1 x2 ↦ x1 ≤ x2) m\np : ι → Set (Set Ω) := fun n ↦ {t |... | exact fun n => (h_indep n).indepSets | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 704,
"column": 45
} | {
"line": 704,
"column": 48
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\na : ι\nS : Finset ι\nhp_ind : iIndepSets π κ μ\nhaS : a ∉ S\nt1 t2 : Set Ω\ns : Finset ι\nhs_mem : s ⊆ S\nft1 : ι → Set Ω\nhft1_mem : ∀ x ∈ s, ft1 x ∈ ... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 705,
"column": 51
} | {
"line": 705,
"column": 54
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\na : ι\nS : Finset ι\nhp_ind : iIndepSets π κ μ\nhaS : a ∉ S\nt1 t2 : Set Ω\ns : Finset ι\nhs_mem : s ⊆ S\nft1 : ι → Set Ω\nhft1_mem : ∀ x ∈ s, ft1 x ∈ ... | ha1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 819,
"column": 17
} | {
"line": 819,
"column": 30
} | [
{
"pp": "case refine_2.hC\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns✝ t : Set Ω\nhs✝ : MeasurableSet s✝\nht : MeasurableSet t\ns' t' : Set Ω\nhs' : s' ∈ {s | MeasurableSet s}\ns : Set Ω\nhs : s ∈ {t}\nht✝ : Meas... | exact hs ▸ ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 819,
"column": 17
} | {
"line": 819,
"column": 30
} | [
{
"pp": "case refine_2.hC\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns✝ t : Set Ω\nhs✝ : MeasurableSet s✝\nht : MeasurableSet t\ns' t' : Set Ω\nhs' : s' ∈ {s | MeasurableSet s}\ns : Set Ω\nhs : s ∈ {t}\nht✝ : Meas... | exact hs ▸ ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 819,
"column": 17
} | {
"line": 819,
"column": 30
} | [
{
"pp": "case refine_2.hC\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns✝ t : Set Ω\nhs✝ : MeasurableSet s✝\nht : MeasurableSet t\ns' t' : Set Ω\nhs' : s' ∈ {s | MeasurableSet s}\ns : Set Ω\nhs : s ∈ {t}\nht✝ : Meas... | exact hs ▸ ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.HasLaw | {
"line": 47,
"column": 15
} | {
"line": 47,
"column": 43
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nX Y : Ω → 𝓧\nμ : Measure 𝓧\nP : Measure Ω\nhX : HasLaw X μ P\nhY : Y =ᶠ[ae P] X\n⊢ map Y P = μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
"ProbabilityTheory.... | rw [map_congr hY, hX.map_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.HasLaw | {
"line": 47,
"column": 15
} | {
"line": 47,
"column": 43
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nX Y : Ω → 𝓧\nμ : Measure 𝓧\nP : Measure Ω\nhX : HasLaw X μ P\nhY : Y =ᶠ[ae P] X\n⊢ map Y P = μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
"ProbabilityTheory.... | rw [map_congr hY, hX.map_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.HasLaw | {
"line": 47,
"column": 15
} | {
"line": 47,
"column": 43
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nX Y : Ω → 𝓧\nμ : Measure 𝓧\nP : Measure Ω\nhX : HasLaw X μ P\nhY : Y =ᶠ[ae P] X\n⊢ map Y P = μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
"ProbabilityTheory.... | rw [map_congr hY, hX.map_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.IdentDistrib | {
"line": 324,
"column": 18
} | {
"line": 324,
"column": 92
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁴ : MeasurableSpace α\nE : Type u_5\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : BorelSpace E\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nι : Type u_6\nf : ι → α → E\nj : ι\np : ℝ≥0∞\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhℒp : MemLp (f j) p μ\nhfmeas : ∀ (i : ι),... | ← eLpNorm_map_measure F_meas.aestronglyMeasurable (hf i).aemeasurable_fst, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Variance | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 47
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : AEStronglyMeasurable X μ\nc : ℝ\n⊢ Var[fun ω ↦ c + X ω; μ] = Var[X; μ]",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.va... | simp_rw [add_comm c, variance_add_const hX c] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Moments.Variance | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 47
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : AEStronglyMeasurable X μ\nc : ℝ\n⊢ Var[fun ω ↦ c + X ω; μ] = Var[X; μ]",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.va... | simp_rw [add_comm c, variance_add_const hX c] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.Variance | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 47
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : AEStronglyMeasurable X μ\nc : ℝ\n⊢ Var[fun ω ↦ c + X ω; μ] = Var[X; μ]",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.va... | simp_rw [add_comm c, variance_add_const hX c] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Variance | {
"line": 347,
"column": 4
} | {
"line": 348,
"column": 18
} | [
{
"pp": "case neg\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhm : AEStronglyMeasurable X μ\nhX : ¬MemLp X 2 μ\nhint : ¬Integrable X μ\n⊢ ∫ (a : Ω), (X a - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ ≤ ∫ (x : Ω), (X ^ 2) x ∂μ",
"usedConstants": [
"Eq.mpr",
"Inn... | · simp only [integral_undef hint, Pi.pow_apply, sub_zero]
exact le_rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.Variance | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 53
} | [
{
"pp": "case neg\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_3\nX : ι → Ω → ℝ\ns : Finset ι\nhs : ∀ i ∈ s, MemLp (X i) 2 μ\nh : (↑s).Pairwise fun i j ↦ X i ⟂ᵢ[μ] X j\nh'' : ¬∀ i ∈ s, X i =ᶠ[ae μ] 0\nj : ι\nhj1 : j ∈ s\nhj2 : ¬X j =ᶠ[ae μ] 0\n⊢ Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, Var[X i; μ]",
... | obtain rfl | h' := s.eq_singleton_or_nontrivial hj1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 70
} | [
{
"pp": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\... | rw [mem_setOf, indicator_of_mem hx.1, eLpNormEssSup_eq_zero_iff.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Process.Filtration | {
"line": 271,
"column": 6
} | {
"line": 278,
"column": 24
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis : TopologicalSpace ι := Preorder.topology ι\ni j : ι\nhij : i ≤ j\n⊢ (fun i ↦ if (𝓝[>] i).NeBot then ⨅ j, ⨅ (_ : j > i), ↑𝓕 j else ↑𝓕 i) i ≤\n (fun i ↦ if (𝓝[>] i).NeBot then ⨅ j, ⨅ (_ : j > i), ... | simp only [gt_iff_lt]
split_ifs with hi hj hj
· exact le_iInf₂ fun k hkj ↦ iInf₂_le k (hij.trans_lt hkj)
· obtain rfl | hj := eq_or_ne j i
· contradiction
· exact iInf₂_le j (lt_of_le_of_ne hij hj.symm)
· exact le_iInf₂ fun k hk ↦ 𝓕.mono (hij.trans hk.le)
· exact 𝓕.mono h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.Filtration | {
"line": 271,
"column": 6
} | {
"line": 278,
"column": 24
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis : TopologicalSpace ι := Preorder.topology ι\ni j : ι\nhij : i ≤ j\n⊢ (fun i ↦ if (𝓝[>] i).NeBot then ⨅ j, ⨅ (_ : j > i), ↑𝓕 j else ↑𝓕 i) i ≤\n (fun i ↦ if (𝓝[>] i).NeBot then ⨅ j, ⨅ (_ : j > i), ... | simp only [gt_iff_lt]
split_ifs with hi hj hj
· exact le_iInf₂ fun k hkj ↦ iInf₂_le k (hij.trans_lt hkj)
· obtain rfl | hj := eq_or_ne j i
· contradiction
· exact iInf₂_le j (lt_of_le_of_ne hij hj.symm)
· exact le_iInf₂ fun k hk ↦ 𝓕.mono (hij.trans hk.le)
· exact 𝓕.mono h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Filtration | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 72
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\n⊢ ⨅ j, ⨅ (_ : j > i), ↑�... | have hle₁ : (⨅ j > i, 𝓕₊ j) ≤ 𝓕₊ v := iInf₂_le_of_le v hv.1 le_rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 478,
"column": 4
} | {
"line": 478,
"column": 42
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k : ι\nhik : ... | filter_upwards [hi, hj] with ω hωi hωj | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 42
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k l : ι\nhik ... | filter_upwards [hi, hj] with ω hωi hωj | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 127,
"column": 80
} | {
"line": 133,
"column": 27
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nr : ℝ≥0∞\ninst✝ : Infinite ι\ns : ι → Set α\nhs : ∀ (i : ι), MeasurableSet (s i)\nhr₀ : r ≠ 0\nhr : ∀ (i : ι), r ≤ μ (s i)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ι⦄, u ⊆ t → u.Finite → 0 < μ (⋂ i ∈ u, s i)",
... | by
obtain ⟨t, ht, h⟩ := bergelson' (fun n ↦ hs <| Infinite.natEmbedding _ n) hr₀ (fun n ↦ hr _)
refine ⟨_, ht.image <| (Infinite.natEmbedding _).injective.injOn, fun u hut hu ↦
(h (preimage_subset_of_surjOn (Infinite.natEmbedding _).injective hut) <| hu.preimage
(Embedding.injective _).injOn).trans_le <| me... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SpecialFunctions.RCLike | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 75
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : MeasurableSpace α\nf : α → 𝕜\nμ : MeasureTheory.Measure α\nhre : AEMeasurable (fun x ↦ RCLike.re (f x)) μ\nhim : AEMeasurable (fun x ↦ RCLike.im (f x)) μ\n⊢ AEMeasurable f μ",
"usedConstants": [
"RCLike.measurable_ofReal",
"Eq.mp... | convert AEMeasurable.add (M := 𝕜) (RCLike.measurable_ofReal.comp_aemeasurable hre)
((RCLike.measurable_ofReal.comp_aemeasurable him).mul_const RCLike.I) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 55,
"column": 8
} | {
"line": 55,
"column": 17
} | [
{
"pp": "p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ (b ^ p⁻¹)⁻¹ • ∫ (x : ℝ) in Ioi (b ^ p⁻¹ * 0), b ^ (-p⁻¹ * q) * (x ^ q * rexp (-x ^ p)) =\n (b ^ p⁻¹)⁻¹ * ∫ (x : ℝ) in Ioi 0, b ^ (-p⁻¹ * q) * (x ^ q * rexp (-x ^ p))",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 706,
"column": 2
} | {
"line": 707,
"column": 39
} | [
{
"pp": "case h.a\nα : Type u_2\ninst✝ : TopologicalSpace α\nf : α →C_c ℝ\na : ℝ\nha : 0 ≤ a\nx : α\n⊢ ↑((a • f).nnrealPart x) = ↑((a.toNNReal • f.nnrealPart) x)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"in... | simp only [nnrealPart_apply, coe_smul, Pi.smul_apply, Real.coe_toNNReal', smul_eq_mul,
NNReal.coe_mul, ha, sup_of_le_left] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 225,
"column": 45
} | {
"line": 225,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b c : E\nt : ℝ\nω : E → E →L[𝕜] F\nγab : Path a b\nγbc : Path b c\nht : t < 1 / 2\ninstE : NormedSpace ℝ E := NormedSpace.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 227,
"column": 4
} | {
"line": 228,
"column": 33
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b c : E\nt : ℝ\nω : E → E →L[𝕜] F\nγab : Path a b\nγbc : Path b c\nht : t < 1 / 2\ninstE : NormedSpace ℝ E := No... | · rw [notMem_closure_iff_nhdsWithin_eq_bot.mp, notMem_closure_iff_nhdsWithin_eq_bot.mp] <;>
simp_intro h <;> linarith | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.LebesgueDifferentiationThm | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 82
} | [
{
"pp": "case h.refine_1\nf : ℝ → ℝ\na b : ℝ\nhf : IntervalIntegrable f volume a b\nhab : a ≤ b\nh₁ : ∀ᵐ (x : ℝ), x ≠ a\nh₂ : ∀ᵐ (x : ℝ), x ≠ b\ng : ℝ → ℝ := fun x ↦ if x ∈ Ioc a b then f x else 0\nhg : LocallyIntegrable g volume\nx : ℝ\nhx : ∀ (c : ℝ), HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, g t) (g x) x\na✝² :... | all_goals apply intervalIntegral.integral_congr_ae' <;> filter_upwards <;> grind | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 40
} | [
{
"pp": "case left\nf : ℝ → ℝ\na b : ℝ\nhf : MonotoneOn f (Icc a b)\nhab : a ≤ b\nG : ℕ → ℝ → ℝ\nhGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))\nhG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))\nhG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop... | filter_upwards [h₁, h₂] with x _ _ _ | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 29
} | [
{
"pp": "case neg.inr\na b : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh : ¬a = b\nthis :\n ∀ {a b : ℝ},\n ContinuousOn f [[a, b]] →\n IntervalIntegrable g μ a b →\n (∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b)... | simp only [not_lt] at hab | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.MeanValue | {
"line": 102,
"column": 4
} | {
"line": 110,
"column": 28
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\ns : Set α\nf g : α → ℝ\nμ : Measure α\nhs_conn : IsConnected s\nhs_meas : MeasurableSet s\nhf : ContinuousOn f s\nhg : IntegrableOn g s μ\nhfg : IntegrableOn (fun x ↦ f x * g x) s μ\nhg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x\... | calc
_ = ∫ x in s, f x ∂ν := heq
_ = f c * ∫ x in s, (1 : ℝ) ∂ν := by
rw [h_ave]
simp only [setAverage_eq, smul_eq_mul, integral_const, MeasurableSet.univ,
measureReal_restrict_apply, Set.univ_inter, mul_one]
rw [measureReal_def]
field_simp
_ = _ := by simp [h... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Integral.Regular | {
"line": 34,
"column": 19
} | {
"line": 34,
"column": 25
} | [
{
"pp": "case a\nX : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\nk : Set X\nhk : IsCompact k\nμ : Measure X\ninst✝³ : IsFiniteMeasureOnCompacts μ\ninst✝² : μ.InnerRegularCompactLTTop\ninst✝¹ : LocallyCompactSpace X\ninst✝ : RegularSpace X\nf : X → ℝ\nf_cont : Contin... | f_comp | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Regular | {
"line": 70,
"column": 6
} | {
"line": 72,
"column": 17
} | [
{
"pp": "case hs\nX : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\ninst✝³ : T2Space X\nU : Set X\nhU : IsOpen U\nμ : Measure X\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : μ.InnerRegularCompactLTTop\ninst✝ : NormalSpace X\nr : ENNReal\nhr : r < μ U\nK : Set X\nKU : K ⊆ U\nK... | intro x hx
apply Eq.ge
exact fK hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Regular | {
"line": 70,
"column": 6
} | {
"line": 72,
"column": 17
} | [
{
"pp": "case hs\nX : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\ninst✝³ : T2Space X\nU : Set X\nhU : IsOpen U\nμ : Measure X\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : μ.InnerRegularCompactLTTop\ninst✝ : NormalSpace X\nr : ENNReal\nhr : r < μ U\nK : Set X\nKU : K ⊆ U\nK... | intro x hx
apply Eq.ge
exact fK hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 84
} | [
{
"pp": "f : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\n⊢ h / 2 *\n (f (a + ↑0 * h) + f (a + (↑(N - 1) + 1) * h) +\n ∑ x ∈ Finset.range (N - 1), (f (a + ↑(x + 1) * h) + f (a + (↑x + 1) * h))) =\n h * ((f a + f (a + ↑N * h)) / 2 + ∑ x ∈ Finset.range (N - 1), f (a + (↑x + 1) * h))",
"usedConstan... | simp_rw [Nat.cast_sub N_nonzero, Nat.cast_add, Nat.cast_one, ← two_mul, ← mul_sum] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Order.Interval.Set.Union | {
"line": 29,
"column": 17
} | {
"line": 32,
"column": 69
} | [
{
"pp": "case succ\nX : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))\n⊢ Ioc (a 0) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ioc (a i) (a (i + 1))",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Finset.mem_range._simp_1",
... | calc
_ ⊆ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) := Ioc_subset_Ioc_union_Ioc
_ ⊆ _ := by simpa [Finset.range_add_one] using
union_subset_union_right (Ioc (a N) (a (N + 1))) ih | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
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