module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 70
} | {
"line": 374,
"column": 71
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrystallographic\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 19
} | {
"line": 372,
"column": 4
} | [
{
"pp": "case a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ni : ι\nx : Set ((i : ι) → α i)\n⊢ ∀ (x_1 : (i : ι) → Set (α i)),\n (∀ (i : ι), MeasurableSet (x_1 i)) → eval i ⁻¹' x_1 i = x → ∃ s S, MeasurableSet S ∧ x = cylinder s S",
"ppTerm": "?a✝",
"assigned": true,
"... | [
"case a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ni : ι\nt : (i : ι) → Set (α i)\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∃ s S, MeasurableSet S ∧ eval i ⁻¹' t i = cylinder s S"
] | rintro t ht rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 376,
"column": 70
} | {
"line": 376,
"column": 81
} | {
"line": 376,
"column": 82
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrystallographic\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 412,
"column": 2
} | {
"line": 412,
"column": 13
} | {
"line": 412,
"column": 14
} | [
{
"pp": "ι : Type u_2\nX : ι → Type u_3\nm : (i : ι) → MeasurableSpace (X i)\nΔ : Set ι\n⊢ cylinderEvents Δ ≤ MeasurableSpace.pi",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\nX : ι → Type u_3\nm : (i : ι) → MeasurableSpace (X i)\nΔ : Set ι\n⊢ cylinderEvents Δ ≤ MeasurableSpace.pi"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 387,
"column": 34
} | {
"line": 387,
"column": 49
} | {
"line": 387,
"column": 50
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrystallographic\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 57
} | {
"line": 127,
"column": 58
} | [
{
"pp": "case convert_2\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝¹ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → Set α\nhf : ∀ (i : ι), f i ∈ C\nh_dis : Pairwise (Disjoint on f)\nh_mem : ⋃ i, f i ∈ C\n⊢ (↑Finset.univ).PairwiseDisjoint f",
"ppTerm": "?convert_2",
... | [
"case convert_2\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝¹ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → Set α\nhf : ∀ (i : ι), f i ∈ C\nh_dis : Pairwise (Disjoint on f)\nh_mem : ⋃ i, f i ∈ C\n⊢ Pairwise (Disjoint on f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 111,
"column": 12
} | {
"line": 111,
"column": 23
} | {
"line": 111,
"column": 24
} | [
{
"pp": "case a.inl.coe.top.inr.refine_1\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... | [
"case a.inl.coe.top.inr.refine_1\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 :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 120,
"column": 62
} | {
"line": 120,
"column": 73
} | {
"line": 120,
"column": 74
} | [
{
"pp": "ι : 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 := if h : ∃ x, Ioi x = ... | [
"ι : 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 := if h : ∃ x, Ioi x = ∅ then h.cho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 229,
"column": 6
} | {
"line": 229,
"column": 17
} | {
"line": 229,
"column": 18
} | [
{
"pp": "α : Type u_1\n𝒜 : Set (Set α)\nh : 𝒜.Countable\nℬ : Set (Set α) := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}\ns : Set α\n⊢ s ∈ compl '' 𝒜 ↔ s ∈ {s | sᶜ ∈ 𝒜}",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.mem_image._simp_1",... | [
"α : Type u_1\n𝒜 : Set (Set α)\nh : 𝒜.Countable\nℬ : Set (Set α) := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}\ns : Set α\n⊢ (∃ x ∈ 𝒜, xᶜ = s) ↔ sᶜ ∈ 𝒜"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 127,
"column": 12
} | {
"line": 127,
"column": 23
} | {
"line": 127,
"column": 24
} | [
{
"pp": "case a.inl.coe.coe.refine_1.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... | [
"case a.inl.coe.coe.refine_1.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 :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 76,
"column": 60
} | {
"line": 76,
"column": 75
} | {
"line": 76,
"column": 76
} | [
{
"pp": "α : 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[PseudoMetricSpace.toUniformSpace... | [
"α : 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[PseudoMetricSpace.toUniformSpace.toTopologic... | iInf_eq_iInter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 256,
"column": 35
} | {
"line": 256,
"column": 50
} | {
"line": 256,
"column": 51
} | [
{
"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\nJ : (s : Set α) → Finpartition s... | [
"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\nJ : (s : Set α) → Finpartition s\nhJC : ∀ s ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 36
} | {
"line": 297,
"column": 37
} | [
{
"pp": "case refine_4\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝² : AddCommMonoid G\nm : AddContent G C\ninst✝¹ : PartialOrder G\ninst✝ : CanonicallyOrderedAdd G\nhC : IsSetSemiring C\nhs : s ∈ C\nht : t ∈ C\nhst : s ⊆ t\nh : ∑ u ∈ {s}, m u ≤ m t\n⊢ m s ≤ m t",
"ppTerm": "?refine_4",
... | [
"case refine_4\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝² : AddCommMonoid G\nm : AddContent G C\ninst✝¹ : PartialOrder G\ninst✝ : CanonicallyOrderedAdd G\nhC : IsSetSemiring C\nhs : s ∈ C\nht : t ∈ C\nhst : s ⊆ t\nh : ∑ u ∈ {s}, m u ≤ m t\n⊢ m s ≤ m t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 160,
"column": 72
} | {
"line": 160,
"column": 83
} | {
"line": 160,
"column": 84
} | [
{
"pp": "ι : 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 := if h : ∃ x, Ioi x = ... | [
"ι : 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 := if h : ∃ x, Ioi x = ∅ then h.cho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 167,
"column": 12
} | {
"line": 167,
"column": 23
} | {
"line": 167,
"column": 24
} | [
{
"pp": "case a.inr.coe.coe.refine_1.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... | [
"case a.inr.coe.coe.refine_1.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 :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 17
} | {
"line": 102,
"column": 18
} | [
{
"pp": "case inr\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[PseudoMetricSpace.toUn... | [
"case inr\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[PseudoMetricSpace.toUniformSpace.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 47
} | {
"line": 113,
"column": 8
} | [
{
"pp": "α : 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[PseudoMetricSpace.toUniformSpace... | [
"α : 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[PseudoMetricSpace.toUniformSpace.toTopologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 429,
"column": 48
} | {
"line": 429,
"column": 59
} | {
"line": 429,
"column": 60
} | [
{
"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)... | [
"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).PairwiseDis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 15
} | {
"line": 280,
"column": 16
} | [
{
"pp": "case inl\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 : IsEmpty ι\n⊢ Tendsto x f (𝓝 ⊤) ↔ ∀ (i : ι), ∀ᶠ (a : α) in f, ↑i < x a",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Pur... | [
"case inl\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 : IsEmpty ι\n⊢ ∀ᶠ (x_1 : α) in f, x x_1 = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 15
} | {
"line": 288,
"column": 16
} | [
{
"pp": "case inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : NoMaxOrder ι\nh : IsEmpty ι\n⊢ Tendsto some atTop (𝓝 ⊤)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"False",
"WithTop.i... | [
"case inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : NoMaxOrder ι\nh : IsEmpty ι\n⊢ atTop = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 54
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case refine_2\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\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 54
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case refine_2\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\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 54
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case refine_2\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\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 49
} | {
"line": 105,
"column": 6
} | [
{
"pp": "case pos\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\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝¹ : SigmaFinite (μ.trim hm)\nthis✝ : s.indicator μ[... | [
"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\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝¹ : SigmaFinite (μ.trim hm)\nthis✝ : s.indicator μ[f | m] =ᵐ[μ]... | · simp only [hx, hxs, Set.indicator_of_mem] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 52
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case refine_2\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\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 52
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case refine_2\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\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 52
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case refine_2\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\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 121,
"column": 71
} | {
"line": 121,
"column": 86
} | {
"line": 121,
"column": 87
} | [
{
"pp": "case refine_1\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\nthis : SigmaFinite ... | [
"case refine_1\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\nthis : SigmaFinite ((μ.restrict... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 166,
"column": 10
} | {
"line": 166,
"column": 25
} | {
"line": 166,
"column": 26
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀... | [
"α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀ (t : Set α)... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 583,
"column": 17
} | {
"line": 583,
"column": 28
} | {
"line": 583,
"column": 29
} | [
{
"pp": "case singleton\nα : Type u_1\nC : Set (Set α)\nι : Type u_2\nhC : IsSetRing C\ns : ι → Set α\nS : Finset ι\na✝ : ι\nhs : ∀ n ∈ {a✝}, s n ∈ C\n⊢ ⋂ i ∈ {a✝}, s i ∈ C",
"ppTerm": "?singleton",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"Set.... | [
"case singleton\nα : Type u_1\nC : Set (Set α)\nι : Type u_2\nhC : IsSetRing C\ns : ι → Set α\nS : Finset ι\na✝ : ι\nhs : ∀ n ∈ {a✝}, s n ∈ C\n⊢ s a✝ ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 593,
"column": 2
} | {
"line": 593,
"column": 13
} | {
"line": 593,
"column": 14
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nhC : IsSetRing C\nι : Type u_2\ns : ι → Set α\nt : Finset ι\nhs : ∀ i ∈ t, s i ∈ C\n⊢ t.sup s ∈ C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteBooleanAlgebra.toCompleteDistribLattice... | [
"α : Type u_1\nC : Set (Set α)\nhC : IsSetRing C\nι : Type u_2\ns : ι → Set α\nt : Finset ι\nhs : ∀ i ∈ t, s i ∈ C\n⊢ ⋃ x ∈ t, s x ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 598,
"column": 2
} | {
"line": 598,
"column": 51
} | {
"line": 598,
"column": 52
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : LocallyFiniteOrderBot ι\nhC : IsSetRing C\ns : ι → Set α\nhs : ∀ (n : ι), s n ∈ C\nn : ι\n⊢ (partialSups s) n ∈ C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeS... | [
"α : Type u_1\nC : Set (Set α)\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : LocallyFiniteOrderBot ι\nhC : IsSetRing C\ns : ι → Set α\nhs : ∀ (n : ι), s n ∈ C\nn : ι\n⊢ (Finset.Iic n).sup s ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 57
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\n⊢ Measurable P⁻[X | mΩ]",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"le_refl",
"MeasurableSpace.instPartialOrder",
"PartialOrder.toPreorder",
"ENNReal.measur... | [] | exact (measurable_condLExp _ _ _).mono hm (le_refl _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 57
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\n⊢ Measurable P⁻[X | mΩ]",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"le_refl",
"MeasurableSpace.instPartialOrder",
"PartialOrder.toPreorder",
"ENNReal.measur... | [] | exact (measurable_condLExp _ _ _).mono hm (le_refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 57
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\n⊢ Measurable P⁻[X | mΩ]",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"le_refl",
"MeasurableSpace.instPartialOrder",
"PartialOrder.toPreorder",
"ENNReal.measur... | [] | exact (measurable_condLExp _ _ _).mono hm (le_refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 35
} | {
"line": 151,
"column": 36
} | [
{
"pp": "Ω : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nhm : mΩ ≤ mΩ₀\nP : Measure Ω\nhσ : SigmaFinite (P.trim hm)\nX : Ω → ℝ≥0∞\n⊢ ∫⁻ (ω : Ω), P⁻[X | mΩ] ω ∂P = ∫⁻ (ω : Ω), X ω ∂P",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
... | [
"Ω : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nhm : mΩ ≤ mΩ₀\nP : Measure Ω\nhσ : SigmaFinite (P.trim hm)\nX : Ω → ℝ≥0∞\n⊢ ∫⁻ (x : Ω) in Set.univ, P⁻[X | mΩ] x ∂P = ∫⁻ (x : Ω) in Set.univ, X x ∂P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 608,
"column": 2
} | {
"line": 615,
"column": 34
} | {
"line": 616,
"column": 2
} | [
{
"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 ... | [
"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 (Disjoint on... | have h_tendsto : Tendsto (fun n ↦ m (s n)) atTop (𝓝 0) := by
refine hm_tendsto hCs ?_ ?_
· intro i j hij x hxj
rw [Set.mem_sdiff] at hxj ⊢
exact ⟨hxj.1, fun hxi ↦ hxj.2 (Set.monotone_accumulate hij hxi)⟩
· simp_rw [s, Set.sdiff_eq]
rw [Set.iInter_inter_distrib, Set.iInter_const, ← Set.com... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 51
} | {
"line": 251,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nα : Type u_2\nm mα : MeasurableSpace α\nμ : Measure α\nf : α → E\nhm : m ≤ mα\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]",
"ppTerm": "?pos✝",
"assi... | [] | simp [condExp_of_not_sigmaFinite hm hμm]; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 51
} | {
"line": 251,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nα : Type u_2\nm mα : MeasurableSpace α\nμ : Measure α\nf : α → E\nhm : m ≤ mα\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]",
"ppTerm": "?pos✝",
"assi... | [] | simp [condExp_of_not_sigmaFinite hm hμm]; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 158,
"column": 91
} | {
"line": 159,
"column": 75
} | {
"line": 160,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nE : Set α\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhδε :\n ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.in... | [] | by
rw [eLpNorm_indicator_eq_eLpNorm_restrict hs, μ.restrict_restrict hs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 17
} | {
"line": 166,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 225,
"column": 8
} | {
"line": 225,
"column": 23
} | {
"line": 225,
"column": 24
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nn m : ℕ\nh✝ : n ≠ m\nh : m < n\n⊢ s.restrictNonposSeq i n ∩ s.restrictNonposSeq i m = ∅",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Set.instIn... | [
"case inr\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nn m : ℕ\nh✝ : n ≠ m\nh : m < n\n⊢ s.restrictNonposSeq i m ∩ s.restrictNonposSeq i n = ∅"
] | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 35
} | {
"line": 163,
"column": 36
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : T2Space M\nv : VectorMeasure α M\nA : Set α\nhA : MeasurableSet A\n⊢ v Aᶜ = v univ - v A",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : T2Space M\nv : VectorMeasure α M\nA : Set α\nhA : MeasurableSet A\n⊢ v (univ \\ A) = v univ - v A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 181,
"column": 50
} | {
"line": 181,
"column": 65
} | {
"line": 181,
"column": 66
} | [
{
"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\nA B : Set α\nhA : MeasurableSet A\nhB : MeasurableSet B\nh' : v (B \\ A) = 0\n⊢ v (A \\ B) + v (B \\ A ∪ A ∩ B) = v (A \\ B) + v B",
"ppTerm": "?m.163"... | [
"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\nA B : Set α\nhA : MeasurableSet A\nhB : MeasurableSet B\nh' : v (B \\ A) = 0\n⊢ v (A \\ B) + v (B \\ A ∪ B ∩ A) = v (A \\ B) + v B"
] | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 250,
"column": 65
} | {
"line": 250,
"column": 76
} | {
"line": 250,
"column": 77
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousSub M\nv : VectorMeasure α M\ns : ℕ → Set α\nhm : Antitone s\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nI : ∀ (n : ℕ), v (s n) = v univ - v (s n)ᶜ\nJ : v (⋂ n, s n) ... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousSub M\nv : VectorMeasure α M\ns : ℕ → Set α\nhm : Antitone s\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nI : ∀ (n : ℕ), v (s n) = v univ - v (s n)ᶜ\nJ : v (⋂ n, s n) = v univ - v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 420,
"column": 40
} | {
"line": 420,
"column": 51
} | {
"line": 420,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : MeasurableSpace β\nx✝ : β\nv✝ : M\ns : Set β\nx : β\nv : M\nf : ℕ → Set β\nf_meas : ∀ (i : ℕ), MeasurableSet (f i)\nf_disj : Pairwise (Disjoint on f)\nhx : x ∈ ⋃ i, f i\nthis ... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : MeasurableSpace β\nx✝ : β\nv✝ : M\ns : Set β\nx : β\nv : M\nf : ℕ → Set β\nf_meas : ∀ (i : ℕ), MeasurableSet (f i)\nf_disj : Pairwise (Disjoint on f)\nhx : x ∈ ⋃ i, f i\nthis : Measurable... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 377,
"column": 8
} | {
"line": 377,
"column": 19
} | {
"line": 377,
"column": 20
} | [
{
"pp": "case neg.h\nα : 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 ≤ EN... | [
"case neg.h\nα : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 377,
"column": 8
} | {
"line": 377,
"column": 19
} | {
"line": 377,
"column": 20
} | [
{
"pp": "case neg.h\nα : 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 ≤ EN... | [
"case neg.h\nα : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 413,
"column": 2
} | {
"line": 419,
"column": 73
} | {
"line": 421,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\n⊢ UnifIntegrable f p μ",
"ppTerm": "?m.20",
"assigned": true,
"use... | [] | intro ε hε
by_cases hι : Nonempty ι
· obtain ⟨i⟩ := hι
obtain ⟨δ, hδpos, hδ⟩ := (hf i).eLpNorm_indicator_le hp_one hp_top hε
refine ⟨δ, hδpos, fun j s hs hμs => ?_⟩
convert! hδ s hs hμs
· exact ⟨1, zero_lt_one, fun i => False.elim <| hι <| Nonempty.intro i⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 413,
"column": 2
} | {
"line": 419,
"column": 73
} | {
"line": 421,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\n⊢ UnifIntegrable f p μ",
"ppTerm": "?m.20",
"assigned": true,
"use... | [] | intro ε hε
by_cases hι : Nonempty ι
· obtain ⟨i⟩ := hι
obtain ⟨δ, hδpos, hδ⟩ := (hf i).eLpNorm_indicator_le hp_one hp_top hε
refine ⟨δ, hδpos, fun j s hs hμs => ?_⟩
convert! hδ s hs hμs
· exact ⟨1, zero_lt_one, fun i => False.elim <| hι <| Nonempty.intro i⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 17
} | {
"line": 451,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\nε : ℝ\nhε : 0 < ε\ng : Fin n → α → β := f ∘ ⇑hn.so... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\nε : ℝ\nhε : 0 < ε\ng : Fin n → α → β := f ∘ ⇑hn.some.symm\nhg ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 398,
"column": 4
} | {
"line": 400,
"column": 12
} | {
"line": 401,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nf : ℕ → ℝ\nleft✝ : Antitone f\nhf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))\nB : ℕ → Set α\nhB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ s (B n) = f n\nhB₁ : ∀ (n : ℕ), MeasurableSet (B n)\nhB₂ : ∀ (n : ℕ), s ≤[B n] 0... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nf : ℕ → ℝ\nleft✝ : Antitone f\nhf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))\nB : ℕ → Set α\nhB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ s (B n) = f n\nhB₁ : ∀ (n : ℕ), MeasurableSet (B n)\nhB₂ : ∀ (n : ℕ), s ≤[B n] 0\nA : Set α ... | rw [← hA₃,
of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC₁) disjoint_compl_right) hA₁
hD₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 85,
"column": 2
} | {
"line": 91,
"column": 25
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g",
"ppTerm": "?m.38",
"assigned": true,
"usedConstan... | [] | ext1 i hi
rw [withDensityᵥ_apply (hf.add hg) hi, _root_.add_apply, withDensityᵥ_apply hf hi,
withDensityᵥ_apply hg hi]
simp_rw [Pi.add_apply]
rw [integral_add]
· exact hf.integrableOn
· exact hg.integrableOn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 85,
"column": 2
} | {
"line": 91,
"column": 25
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g",
"ppTerm": "?m.38",
"assigned": true,
"usedConstan... | [] | ext1 i hi
rw [withDensityᵥ_apply (hf.add hg) hi, _root_.add_apply, withDensityᵥ_apply hf hi,
withDensityᵥ_apply hg hi]
simp_rw [Pi.add_apply]
rw [integral_add]
· exact hf.integrableOn
· exact hg.integrableOn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.RadonNikodym | {
"line": 41,
"column": 6
} | {
"line": 41,
"column": 39
} | {
"line": 42,
"column": 6
} | [
{
"pp": "case hf\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : SigmaFinite μ\nh : s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ\ni : Set α\nhi : MeasurableSet i\n⊢ Integrable (fun x ↦ (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal) (μ.restric... | [
"case hf\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : SigmaFinite μ\nh : s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ\ni : Set α\nhi : MeasurableSet i\n⊢ Integrable (fun x ↦ (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal) μ"
] | refine Integrable.integrableOn ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.RadonNikodym | {
"line": 41,
"column": 6
} | {
"line": 41,
"column": 39
} | {
"line": 42,
"column": 6
} | [
{
"pp": "case hg\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : SigmaFinite μ\nh : s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ\ni : Set α\nhi : MeasurableSet i\n⊢ Integrable (fun x ↦ (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal) (μ.restric... | [
"case hg\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : SigmaFinite μ\nh : s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ\ni : Set α\nhi : MeasurableSet i\n⊢ Integrable (fun x ↦ (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal) μ"
] | refine Integrable.integrableOn ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofRea... | [] | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofRea... | [] | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofRea... | [] | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 47,
"column": 2
} | {
"line": 53,
"column": 84
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhp₀ : p ≠ 0\nhp : p ≠ ∞\nhf : AEStronglyMeasurable f μ\n⊢ lpNorm f p μ = (∫ (x : α), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants":... | [] | rw [← toReal_eLpNorm hf, eLpNorm_eq_lintegral_rpow_enorm_toReal hp₀ hp, ← ENNReal.toReal_rpow,
← integral_toReal]
· simp [← ENNReal.toReal_rpow]
· simp_rw [← ofReal_norm]
borelize E
fun_prop
· exact .of_forall fun x ↦ ENNReal.rpow_lt_top_of_nonneg (by positivity) (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 47,
"column": 2
} | {
"line": 53,
"column": 84
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhp₀ : p ≠ 0\nhp : p ≠ ∞\nhf : AEStronglyMeasurable f μ\n⊢ lpNorm f p μ = (∫ (x : α), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants":... | [] | rw [← toReal_eLpNorm hf, eLpNorm_eq_lintegral_rpow_enorm_toReal hp₀ hp, ← ENNReal.toReal_rpow,
← integral_toReal]
· simp [← ENNReal.toReal_rpow]
· simp_rw [← ofReal_norm]
borelize E
fun_prop
· exact .of_forall fun x ↦ ENNReal.rpow_lt_top_of_nonneg (by positivity) (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 95,
"column": 58
} | {
"line": 95,
"column": 69
} | {
"line": 95,
"column": 70
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable (-f) μ\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable (-f) μ\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 20
} | {
"line": 141,
"column": 21
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nc : 𝕜\nf : α → E\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nhc : c ≠ 0\nh : AEStronglyMeasurable (c • f) μ\n⊢ AEStr... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nc : 𝕜\nf : α → E\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nhc : c ≠ 0\nh : AEStronglyMeasurable (c • f) μ\n⊢ AEStronglyMeasura... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 38
} | {
"line": 146,
"column": 39
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nf : α → E\nμ : Measure α\n⊢ lpNorm (n • f) p μ = n • lpNorm f p μ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toA... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nf : α → E\nμ : Measure α\n⊢ lpNorm (n • f) p μ = ↑n * lpNorm f p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 33
} | {
"line": 152,
"column": 34
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nn : ℕ\nf : α → 𝕜\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (↑n * f) p μ = ↑n * lpNorm f p μ",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nn : ℕ\nf : α → 𝕜\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (↑n * f) p μ = ↑n * lpNorm f p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 29
} | {
"line": 159,
"column": 30
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nf : α → 𝕜\nn : ℕ\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (f * ↑n) p μ = lpNorm f p μ * ↑n",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nf : α → 𝕜\nn : ℕ\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (f * ↑n) p μ = lpNorm f p μ * ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 182,
"column": 47
} | {
"line": 182,
"column": 58
} | {
"line": 182,
"column": 59
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : ¬MemLp g p μ\nhfg : MemLp (f + g) p μ\n⊢ MemLp g p μ",
"ppTerm": "?m.185",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : ¬MemLp g p μ\nhfg : MemLp (f + g) p μ\n⊢ MemLp g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 24
} | {
"line": 187,
"column": 25
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 30
} | {
"line": 191,
"column": 31
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\n⊢ lpNorm (f - g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instL... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\n⊢ lpNorm (f + -g) p μ ≤ lpNorm f p μ + lpNorm g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | {
"line": 195,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (f - g) p μ",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (f - g) p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 30
} | {
"line": 199,
"column": 31
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (g - f) p μ",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (g - f) p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | {
"line": 203,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm (f + g) p μ + lpNorm g p μ",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm (f + g) p μ + lpNorm g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g h : α → E\nhf : MemLp f p μ\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f - h) p μ ≤ lpNorm (f - g) p μ + lpNorm (g - h) p μ",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": []... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g h : α → E\nhf : MemLp f p μ\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f - h) p μ ≤ lpNorm (f - g) p μ + lpNorm (g - h) p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 13
} | {
"line": 215,
"column": 14
} | [
{
"pp": "case hb\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nι : Type u_3\ns : Finset ι\nf : ι → α → E\nhf : ∀ i ∈ s, MemLp (f i) p μ\nhp : 1 ≤ p\n⊢ ∑ i ∈ s, eLpNorm (f i) p μ ≠ ∞",
"ppTerm": "?hb",
"assigned": true,
"usedConstants": [
... | [
"case hb\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nι : Type u_3\ns : Finset ι\nf : ι → α → E\nhf : ∀ i ∈ s, MemLp (f i) p μ\nhp : 1 ≤ p\n⊢ ∀ x ∈ s, ¬eLpNorm (f x) p μ = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 240,
"column": 6
} | {
"line": 240,
"column": 22
} | {
"line": 240,
"column": 23
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nc : ℝ≥0\nhc : c ≠ 0\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.61",
"assigned": false,
"usedConstants": [... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nc : ℝ≥0\nhc : c ≠ 0\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 20
} | {
"line": 254,
"column": 21
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nhp : p ≠ ∞\nf : α → E\nc : ℝ≥0\nhf : ¬AEStronglyMeasurable f μ\nhp₀ : p ≠ 0\nhc : c ≠ 0\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.113",
"assigned": fal... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nhp : p ≠ ∞\nf : α → E\nc : ℝ≥0\nhf : ¬AEStronglyMeasurable f μ\nhp₀ : p ≠ 0\nhc : c ≠ 0\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 282,
"column": 2
} | {
"line": 284,
"column": 72
} | {
"line": 285,
"column": 2
} | [
{
"pp": "case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : SignedMeasure α\nf : α → ℝ\nhf : Measurable f\nhfi : Integrable f μ\nhtμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ\nhadd : s = t + μ.withDensityᵥ f\nhtμ' : t ⟂ᵥ μ.toENNRealVectorMeasure\n⊢ t.toJordan... | [
"case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : SignedMeasure α\nf : α → ℝ\nhf : Measurable f\nhfi : Integrable f μ\nhtμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ\nhadd : s = t + μ.withDensityᵥ f\nhtμ' : t ⟂ᵥ μ.toENNRealVectorMeasure\n⊢ t.toJordanDecompositio... | · have hfpos : Measurable fun x => ENNReal.ofReal (f x) := by fun_prop
refine eq_singularPart hfpos htμ.1 ?_
rw [toJordanDecomposition_eq_of_eq_add_withDensity hf hfi htμ' hadd] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"ppTerm": "?pos.hf✝",
"assigned": true,
"usedConstants": [
"AEMeasurable.aestrongly... | [] | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"ppTerm": "?pos.hf✝",
"assigned": true,
"usedConstants": [
"AEMeasurable.aestrongly... | [] | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"ppTerm": "?pos.hf✝",
"assigned": true,
"usedConstants": [
"AEMeasurable.aestrongly... | [] | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 264,
"column": 4
} | {
"line": 265,
"column": 11
} | {
"line": 265,
"column": 12
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable ((starRingEnd (α → K)) f) μ\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.106",
"assigned": false,
"usedConstants": [],
"usedFV... | [
"α : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable ((starRingEnd (α → K)) f) μ\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 395,
"column": 2
} | {
"line": 399,
"column": 44
} | {
"line": 401,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns t : SignedMeasure α\nμ : Measure α\ninst✝² : s.HaveLebesgueDecomposition μ\ninst✝¹ : t.HaveLebesgueDecomposition μ\ninst✝ : (s + t).HaveLebesgueDecomposition μ\n⊢ μ.withDensityᵥ ((s + t).rnDeriv μ) = μ.withDensityᵥ (s.rnDeriv μ + t.rnDeriv μ)",
"ppTerm": "?m.6... | [] | rw [← add_right_inj ((s + t).singularPart μ), singularPart_add_withDensity_rnDeriv_eq,
withDensityᵥ_add (integrable_rnDeriv _ _) (integrable_rnDeriv _ _), singularPart_add,
add_assoc, add_comm (t.singularPart μ), add_assoc, add_comm _ (t.singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, ← add_assoc,... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 62
} | {
"line": 69,
"column": 63
} | [
{
"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) ... | [
"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) ω ≤ c\nf : Ω... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 63
} | {
"line": 70,
"column": 64
} | [
{
"pp": "case refine_2\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) ... | [
"case refine_2\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) ω ≤ c\nf : Ω... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | {
"line": 129,
"column": 14
} | [
{
"pp": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nδ : ℝ\nδ_pos : 0 < δ\nx : Ω\n⊢ ‖↑((thickenedIndicator δ_pos F) x)‖ ≤ 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"... | [
"Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nδ : ℝ\nδ_pos : 0 < δ\nx : Ω\n⊢ (thickenedIndicatorAux δ F x).toNNReal ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 33
} | {
"line": 143,
"column": 33
} | [
{
"pp": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝... | [
"Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nfs : ℕ ... | lintegral_coe_eq_integral | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 38
} | {
"line": 95,
"column": 39
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : ¬m ≤ m0\n⊢ ∫ (x : α)... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : ¬m ≤ m0\n⊢ 0 ≤ ∫ (x : α), |f x| ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 52
} | {
"line": 97,
"column": 53
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : m ≤ m0\nhsig : ¬Sigm... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : m ≤ m0\nhsig : ¬SigmaFinite (μ.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 38
} | {
"line": 117,
"column": 39
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf :... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 49
} | {
"line": 119,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf :... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 52
} | {
"line": 121,
"column": 53
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf :... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 38
} | {
"line": 132,
"column": 39
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : ¬m ≤ m0\n⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inn... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : ¬m ≤ m0\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 49
} | {
"line": 134,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : m ≤ m0\nhfint : ¬Integrable f μ\n⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : m ≤ m0\nhfint : ¬Integrable f μ\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 52
} | {
"line": 136,
"column": 53
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : m ≤ m0\nhfint : Integrable f μ\nhsig : ¬SigmaFinite (μ.trim hm)\n⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ",
"ppTerm": "?pos✝",
"assigned":... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : m ≤ m0\nhfint : Integrable f μ\nhsig : ¬SigmaFinite (μ.trim hm)\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 49
} | {
"line": 208,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : ¬Integrable f μ\n⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ",
"ppTerm": "?pos✝",
"assigned": true,
"use... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : ¬Integrable f μ\n⊢ 0 ≤ ∫ (x : α), ‖f x‖ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 13
} | {
"line": 209,
"column": 14
} | [
{
"pp": "case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : Integrable f μ\n⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ",
"ppTerm": "?neg✝",
"assigned": false,
"use... | [
"case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : Integrable f μ\n⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 209,
"column": 55
} | {
"line": 209,
"column": 66
} | {
"line": 209,
"column": 67
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : Integrable f μ\n⊢ Integrable (fun x ↦ ‖f x‖ ^ 1) μ",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Norm.... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : Integrable f μ\n⊢ Integrable (fun x ↦ ‖f x‖) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 66
} | {
"line": 138,
"column": 67
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\ns✝ t : Set Ω\nμ ν : FiniteMeasure Ω\nh : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν\ns : Set Ω\nx✝ : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\ns✝ t : Set Ω\nμ ν : FiniteMeasure Ω\nh : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν\ns : Set Ω\nx✝ : MeasurableSet s\n⊢ ↑μ s = ↑ν s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 186,
"column": 2
} | {
"line": 187,
"column": 9
} | {
"line": 187,
"column": 10
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\nμ : FiniteMeasure Ω\nf : ι → Set Ω\n⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (μ (⋃ i, f i)))",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\nμ : FiniteMeasure Ω\nf : ι → Set Ω\n⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (μ (⋃ i, f i)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 43
} | {
"line": 216,
"column": 44
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : ¬m ≤ m0\n⊢ ∫ (x : α) i... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : ¬m ≤ m0\n⊢ 0 ≤ ∫ (x : α) in s, ‖f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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