full_name
stringlengths 3
121
| state
stringlengths 7
9.32k
| tactic
stringlengths 3
5.35k
| target_state
stringlengths 7
19k
| url
stringclasses 1
value | commit
stringclasses 1
value | file_path
stringlengths 21
79
|
|---|---|---|---|---|---|---|
CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_hom_comp_i
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology
⊢ h.cyclesIso.hom ≫ h.i = S.iCycles
|
dsimp [<a>CategoryTheory.ShortComplex.iCycles</a>, <a>CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso</a>]
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology
⊢ cyclesMap' (𝟙 S) S.leftHomologyData h ≫ h.i = S.leftHomologyData.i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
|
CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_hom_comp_i
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology
⊢ cyclesMap' (𝟙 S) S.leftHomologyData h ≫ h.i = S.leftHomologyData.i
|
simp only [<a>CategoryTheory.ShortComplex.cyclesMap'_i</a>, <a>CategoryTheory.ShortComplex.id_τ₂</a>, <a>CategoryTheory.Category.comp_id</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
|
Cardinal.ord_le
|
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
⊢ (#α).ord ≤ type s ↔ #α ≤ (type s).card
|
let ⟨r, _, e⟩ := <a>Cardinal.ord_eq</a> α
|
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
⊢ (#α).ord ≤ type s ↔ #α ≤ (type s).card
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.ord_le
|
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
⊢ (#α).ord ≤ type s ↔ #α ≤ (type s).card
|
simp only [<a>Ordinal.card_type</a>]
|
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
⊢ (#α).ord ≤ type s ↔ #α ≤ #β
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.ord_le
|
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
⊢ (#α).ord ≤ type s ↔ #α ≤ #β
|
constructor <;> intro h
|
case mp
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
h : (#α).ord ≤ type s
⊢ #α ≤ #β
case mpr
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
h : #α ≤ #β
⊢ (#α).ord ≤ type s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.ord_le
|
case mp
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
h : (#α).ord ≤ type s
⊢ #α ≤ #β
|
rw [e] at h
|
case mp
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
h : type r ≤ type s
⊢ #α ≤ #β
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.ord_le
|
case mp
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
h : type r ≤ type s
⊢ #α ≤ #β
|
exact let ⟨f⟩ := h ⟨f.toEmbedding⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.ord_le
|
case mpr
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
h : #α ≤ #β
⊢ (#α).ord ≤ type s
|
cases' h with f
|
case mpr.intro
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
f : α ↪ β
⊢ (#α).ord ≤ type s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.ord_le
|
case mpr.intro
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
f : α ↪ β
⊢ (#α).ord ≤ type s
|
have g := <a>RelEmbedding.preimage</a> f s
|
case mpr.intro
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
f : α ↪ β
g : ⇑f ⁻¹'o s ↪r s
⊢ (#α).ord ≤ type s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.ord_le
|
case mpr.intro
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
f : α ↪ β
g : ⇑f ⁻¹'o s ↪r s
⊢ (#α).ord ≤ type s
|
haveI := <a>RelEmbedding.isWellOrder</a> g
|
case mpr.intro
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
f : α ↪ β
g : ⇑f ⁻¹'o s ↪r s
this : IsWellOrder α (⇑f ⁻¹'o s)
⊢ (#α).ord ≤ type s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.ord_le
|
case mpr.intro
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
c : Cardinal.{u_3}
o : Ordinal.{u_3}
α β : Type u_3
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
f : α ↪ β
g : ⇑f ⁻¹'o s ↪r s
this : IsWellOrder α (⇑f ⁻¹'o s)
⊢ (#α).ord ≤ type s
|
exact <a>le_trans</a> (<a>Cardinal.ord_le_type</a> _) g.ordinal_type_le
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
Primrec.list_length
|
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
l : List α
⊢ List.foldr (fun b s => (fun a b => (a, b).2.2.succ) (l, b, s).1 (l, b, s).2) 0 (id l) = l.length
|
dsimp
|
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
l : List α
⊢ List.foldr (fun b s => s + 1) 0 l = l.length
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Computability/Primrec.lean
|
Primrec.list_length
|
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
l : List α
⊢ List.foldr (fun b s => s + 1) 0 l = l.length
|
induction l <;> simp [*]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Computability/Primrec.lean
|
MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
⊢ MeasurePreserving (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ) (μ.toSphere.prod (volumeIoiPow (dim E - 1)))
|
nontriviality E
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
⊢ MeasurePreserving (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ) (μ.toSphere.prod (volumeIoiPow (dim E - 1)))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
|
MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
⊢ MeasurePreserving (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ) (μ.toSphere.prod (volumeIoiPow (dim E - 1)))
|
refine ⟨(<a>homeomorphUnitSphereProd</a> E).<a>Homeomorph.measurable</a>, .symm ?_⟩
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
⊢ μ.toSphere.prod (volumeIoiPow (dim E - 1)) = map (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
|
MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
⊢ μ.toSphere.prod (volumeIoiPow (dim E - 1)) = map (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ)
|
refine <a>MeasureTheory.Measure.prod_eq_generateFrom</a> <a>MeasurableSpace.generateFrom_measurableSet</a> ((<a>borel_eq_generateFrom_Iio</a> _).symm.trans BorelSpace.measurable_eq.symm) <a>MeasurableSpace.isPiSystem_measurableSet</a> <a>isPiSystem_Iio</a> μ.toSphere.toFiniteSpanningSetsIn (<a>MeasureTheory.Measure.finiteSpanningSetsIn_volumeIoiPow_range_Iio</a> _) fun s hs ↦ <a>Set.forall_mem_range</a>.2 fun r ↦ ?_
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
⊢ (map (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ)) (s ×ˢ Iio r) =
μ.toSphere s * (volumeIoiPow (dim E - 1)) (Iio r)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
|
MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
⊢ (map (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ)) (s ×ˢ Iio r) =
μ.toSphere s * (volumeIoiPow (dim E - 1)) (Iio r)
|
have : <a>Set.Ioo</a> (0 : ℝ) r = r.1 • <a>Set.Ioo</a> (0 : ℝ) 1 := by rw [<a>LinearOrderedField.smul_Ioo</a> r.2.<a>Membership.mem.out</a>, <a>smul_zero</a>, <a>smul_eq_mul</a>, <a>mul_one</a>]
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
⊢ (map (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ)) (s ×ˢ Iio r) =
μ.toSphere s * (volumeIoiPow (dim E - 1)) (Iio r)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
|
MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
⊢ (map (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ)) (s ×ˢ Iio r) =
μ.toSphere s * (volumeIoiPow (dim E - 1)) (Iio r)
|
have hpos : 0 < dim E := <a>FiniteDimensional.finrank_pos</a>
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
hpos : 0 < dim E
⊢ (map (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ)) (s ×ˢ Iio r) =
μ.toSphere s * (volumeIoiPow (dim E - 1)) (Iio r)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
|
MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
hpos : 0 < dim E
⊢ (map (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ)) (s ×ˢ Iio r) =
μ.toSphere s * (volumeIoiPow (dim E - 1)) (Iio r)
|
rw [(<a>Homeomorph.measurableEmbedding</a> _).<a>MeasurableEmbedding.map_apply</a>, <a>MeasureTheory.Measure.toSphere_apply'</a> _ hs, <a>MeasureTheory.Measure.volumeIoiPow_apply_Iio</a>, <a>comap_subtype_coe_apply</a> (<a>MeasurableSingletonClass.measurableSet_singleton</a> _).<a>MeasurableSet.compl</a>, <a>MeasureTheory.Measure.toSphere_apply_aux</a>, this, <a>smul_assoc</a>, μ.addHaar_smul_of_nonneg r.2.out.le, <a>Nat.sub_add_cancel</a> hpos, <a>Nat.cast_pred</a> hpos, <a>sub_add_cancel</a>, <a>mul_right_comm</a>, ← <a>ENNReal.ofReal_natCast</a>, ← <a>ENNReal.ofReal_mul</a>, <a>mul_div_cancel₀</a>]
|
case hb
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
hpos : 0 < dim E
⊢ ↑(dim E) ≠ 0
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
hpos : 0 < dim E
⊢ 0 ≤ ↑(dim E)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
|
MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
|
case hb
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
hpos : 0 < dim E
⊢ ↑(dim E) ≠ 0
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
hpos : 0 < dim E
⊢ 0 ≤ ↑(dim E)
|
exacts [(<a>Nat.cast_pos</a>.2 hpos).<a>LT.lt.ne'</a>, <a>Nat.cast_nonneg</a> _]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
|
MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
⊢ Ioo 0 ↑r = ↑r • Ioo 0 1
|
rw [<a>LinearOrderedField.smul_Ioo</a> r.2.<a>Membership.mem.out</a>, <a>smul_zero</a>, <a>smul_eq_mul</a>, <a>mul_one</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
|
AddCircle.addOrderOf_div_of_gcd_eq_one
|
𝕜 : Type u_1
B : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p q : 𝕜
hp : Fact (0 < p)
m n : ℕ
hn : 0 < n
h : m.gcd n = 1
⊢ addOrderOf ↑(↑m / ↑n * p) = n
|
convert <a>AddCircle.gcd_mul_addOrderOf_div_eq</a> p m hn
|
case h.e'_2
𝕜 : Type u_1
B : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p q : 𝕜
hp : Fact (0 < p)
m n : ℕ
hn : 0 < n
h : m.gcd n = 1
⊢ addOrderOf ↑(↑m / ↑n * p) = m.gcd n * addOrderOf ↑(↑m / ↑n * p)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Instances/AddCircle.lean
|
AddCircle.addOrderOf_div_of_gcd_eq_one
|
case h.e'_2
𝕜 : Type u_1
B : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p q : 𝕜
hp : Fact (0 < p)
m n : ℕ
hn : 0 < n
h : m.gcd n = 1
⊢ addOrderOf ↑(↑m / ↑n * p) = m.gcd n * addOrderOf ↑(↑m / ↑n * p)
|
rw [h, <a>one_mul</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Instances/AddCircle.lean
|
Uniform.continuousAt_iff'_right
|
α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
f : β → α
b : β
⊢ ContinuousAt f b ↔ Tendsto (fun x => (f b, f x)) (𝓝 b) (𝓤 α)
|
rw [<a>ContinuousAt</a>, <a>Uniform.tendsto_nhds_right</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/UniformSpace/Basic.lean
|
Filter.hasBasis_biInf_principal
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s✝ : ι → Set α
t✝ : Set α
i : ι
p' : ι' → Prop
s' : ι' → Set α
i' : ι'
s : β → Set α
S : Set β
h : DirectedOn (s ⁻¹'o fun x x_1 => x ≥ x_1) S
ne : S.Nonempty
t : Set α
⊢ t ∈ ⨅ i ∈ S, 𝓟 (s i) ↔ ∃ i ∈ S, s i ⊆ t
|
refine <a>Filter.mem_biInf_of_directed</a> ?_ ne
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s✝ : ι → Set α
t✝ : Set α
i : ι
p' : ι' → Prop
s' : ι' → Set α
i' : ι'
s : β → Set α
S : Set β
h : DirectedOn (s ⁻¹'o fun x x_1 => x ≥ x_1) S
ne : S.Nonempty
t : Set α
⊢ DirectedOn ((fun i => 𝓟 (s i)) ⁻¹'o fun x x_1 => x ≥ x_1) S
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Bases.lean
|
Filter.hasBasis_biInf_principal
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s✝ : ι → Set α
t✝ : Set α
i : ι
p' : ι' → Prop
s' : ι' → Set α
i' : ι'
s : β → Set α
S : Set β
h : DirectedOn (s ⁻¹'o fun x x_1 => x ≥ x_1) S
ne : S.Nonempty
t : Set α
⊢ DirectedOn ((fun i => 𝓟 (s i)) ⁻¹'o fun x x_1 => x ≥ x_1) S
|
rw [<a>directedOn_iff_directed</a>, ← <a>directed_comp</a>] at h ⊢
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s✝ : ι → Set α
t✝ : Set α
i : ι
p' : ι' → Prop
s' : ι' → Set α
i' : ι'
s : β → Set α
S : Set β
h : Directed (fun x x_1 => x ≥ x_1) (s ∘ Subtype.val)
ne : S.Nonempty
t : Set α
⊢ Directed (fun x x_1 => x ≥ x_1) ((fun i => 𝓟 (s i)) ∘ Subtype.val)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Bases.lean
|
Filter.hasBasis_biInf_principal
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s✝ : ι → Set α
t✝ : Set α
i : ι
p' : ι' → Prop
s' : ι' → Set α
i' : ι'
s : β → Set α
S : Set β
h : Directed (fun x x_1 => x ≥ x_1) (s ∘ Subtype.val)
ne : S.Nonempty
t : Set α
⊢ Directed (fun x x_1 => x ≥ x_1) ((fun i => 𝓟 (s i)) ∘ Subtype.val)
|
refine h.mono_comp ?_
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s✝ : ι → Set α
t✝ : Set α
i : ι
p' : ι' → Prop
s' : ι' → Set α
i' : ι'
s : β → Set α
S : Set β
h : Directed (fun x x_1 => x ≥ x_1) (s ∘ Subtype.val)
ne : S.Nonempty
t : Set α
⊢ ∀ ⦃x y : Set α⦄, x ≥ y → 𝓟 x ≥ 𝓟 y
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Bases.lean
|
Filter.hasBasis_biInf_principal
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s✝ : ι → Set α
t✝ : Set α
i : ι
p' : ι' → Prop
s' : ι' → Set α
i' : ι'
s : β → Set α
S : Set β
h : Directed (fun x x_1 => x ≥ x_1) (s ∘ Subtype.val)
ne : S.Nonempty
t : Set α
⊢ ∀ ⦃x y : Set α⦄, x ≥ y → 𝓟 x ≥ 𝓟 y
|
exact fun _ _ => <a>Filter.principal_mono</a>.2
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Bases.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
|
have huniv : ∀ᵐ x ∂ρ.fst, κ x <a>Set.univ</a> = ρ.condKernel x <a>Set.univ</a> := <a>ProbabilityTheory.eq_condKernel_of_measure_eq_compProd'</a> κ hκ <a>MeasurableSet.univ</a>
|
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
|
suffices ∀ᵐ x ∂ρ.fst, ∀ ⦃t⦄, <a>MeasurableSet</a> t → κ x t = ρ.condKernel x t by filter_upwards [this] with x hx ext t ht; exact hx ht
|
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
|
apply <a>MeasurableSpace.ae_induction_on_inter</a> <a>Real.borel_eq_generateFrom_Iic_rat</a> <a>Real.isPiSystem_Iic_rat</a>
|
case h_empty
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, (κ x) ∅ = (ρ.condKernel x) ∅
case h_basic
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ t ∈ ⋃ a, {Iic ↑a}, (κ x) t = (ρ.condKernel x) t
case h_compl
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (t : Set ℝ), MeasurableSet t → (κ x) t = (ρ.condKernel x) t → (κ x) tᶜ = (ρ.condKernel x) tᶜ
case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst,
∀ (f : ℕ → Set ℝ),
Pairwise (Disjoint on f) →
(∀ (i : ℕ), MeasurableSet (f i)) →
(∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)) → (κ x) (⋃ i, f i) = (ρ.condKernel x) (⋃ i, f i)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
|
filter_upwards [this] with x hx
|
case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
x : α
hx : ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
⊢ κ x = ρ.condKernel x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
x : α
hx : ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
⊢ κ x = ρ.condKernel x
|
ext t ht
|
case h.h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
x : α
hx : ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
t : Set ℝ
ht : MeasurableSet t
⊢ (κ x) t = (ρ.condKernel x) t
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h.h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
x : α
hx : ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
t : Set ℝ
ht : MeasurableSet t
⊢ (κ x) t = (ρ.condKernel x) t
|
exact hx ht
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h_empty
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, (κ x) ∅ = (ρ.condKernel x) ∅
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h_basic
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ t ∈ ⋃ a, {Iic ↑a}, (κ x) t = (ρ.condKernel x) t
|
simp only [<a>Set.iUnion_singleton_eq_range</a>, <a>Set.mem_range</a>, <a>forall_exists_index</a>, <a>forall_apply_eq_imp_iff</a>]
|
case h_basic
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (a : ℚ), (κ x) (Iic ↑a) = (ρ.condKernel x) (Iic ↑a)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h_basic
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (a : ℚ), (κ x) (Iic ↑a) = (ρ.condKernel x) (Iic ↑a)
|
exact <a>MeasureTheory.ae_all_iff</a>.2 fun q ↦ <a>ProbabilityTheory.eq_condKernel_of_measure_eq_compProd'</a> κ hκ <a>measurableSet_Iic</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h_compl
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (t : Set ℝ), MeasurableSet t → (κ x) t = (ρ.condKernel x) t → (κ x) tᶜ = (ρ.condKernel x) tᶜ
|
filter_upwards [huniv] with x hxuniv t ht heq
|
case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
hxuniv : (κ x) univ = (ρ.condKernel x) univ
t : Set ℝ
ht : MeasurableSet t
heq : (κ x) t = (ρ.condKernel x) t
⊢ (κ x) tᶜ = (ρ.condKernel x) tᶜ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
hxuniv : (κ x) univ = (ρ.condKernel x) univ
t : Set ℝ
ht : MeasurableSet t
heq : (κ x) t = (ρ.condKernel x) t
⊢ (κ x) tᶜ = (ρ.condKernel x) tᶜ
|
rw [<a>MeasureTheory.measure_compl</a> ht <| <a>MeasureTheory.measure_ne_top</a> _ _, heq, hxuniv, <a>MeasureTheory.measure_compl</a> ht <| <a>MeasureTheory.measure_ne_top</a> _ _]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst,
∀ (f : ℕ → Set ℝ),
Pairwise (Disjoint on f) →
(∀ (i : ℕ), MeasurableSet (f i)) →
(∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)) → (κ x) (⋃ i, f i) = (ρ.condKernel x) (⋃ i, f i)
|
refine <a>MeasureTheory.ae_of_all</a> _ (fun x f hdisj hf heq ↦ ?_)
|
case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
f : ℕ → Set ℝ
hdisj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
heq : ∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)
⊢ (κ x) (⋃ i, f i) = (ρ.condKernel x) (⋃ i, f i)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
f : ℕ → Set ℝ
hdisj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
heq : ∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)
⊢ (κ x) (⋃ i, f i) = (ρ.condKernel x) (⋃ i, f i)
|
rw [<a>MeasureTheory.measure_iUnion</a> hdisj hf, <a>MeasureTheory.measure_iUnion</a> hdisj hf]
|
case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
f : ℕ → Set ℝ
hdisj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
heq : ∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)
⊢ ∑' (i : ℕ), (κ x) (f i) = ∑' (i : ℕ), (ρ.condKernel x) (f i)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
|
case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
f : ℕ → Set ℝ
hdisj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
heq : ∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)
⊢ ∑' (i : ℕ), (κ x) (f i) = ∑' (i : ℕ), (ρ.condKernel x) (f i)
|
exact <a>tsum_congr</a> heq
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/Unique.lean
|
Polynomial.degree_C_mul
|
R : Type u
S : Type v
ι : Type w
a✝ b : R
m n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q : R[X]
a : R
a0 : a ≠ 0
⊢ (C a * p).degree = p.degree
|
rw [<a>Polynomial.degree_mul</a>, <a>Polynomial.degree_C</a> a0, <a>zero_add</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
|
Decidable.forall_or_right
|
α : Sort u_1
β : Sort u_2
p✝ q✝ : α → Prop
q : Prop
p : α → Prop
inst✝ : Decidable q
⊢ (∀ (x : α), p x ∨ q) ↔ (∀ (x : α), p x) ∨ q
|
simp [<a>or_comm</a>, <a>Decidable.forall_or_left</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Logic/Basic.lean
|
EuclideanGeometry.inversion_dist_center'
|
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c✝ d x✝ y z : P
r R : ℝ
c x : P
⊢ inversion c (dist c x) x = x
|
rw [<a>dist_comm</a>, <a>EuclideanGeometry.inversion_dist_center</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Inversion/Basic.lean
|
Ideal.IsHomogeneous.sInf
|
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
ℐ : Set (Ideal A)
h : ∀ I ∈ ℐ, IsHomogeneous 𝒜 I
⊢ IsHomogeneous 𝒜 (InfSet.sInf ℐ)
|
rw [<a>sInf_eq_iInf</a>]
|
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
ℐ : Set (Ideal A)
h : ∀ I ∈ ℐ, IsHomogeneous 𝒜 I
⊢ IsHomogeneous 𝒜 (⨅ a ∈ ℐ, a)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.IsHomogeneous.sInf
|
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
ℐ : Set (Ideal A)
h : ∀ I ∈ ℐ, IsHomogeneous 𝒜 I
⊢ IsHomogeneous 𝒜 (⨅ a ∈ ℐ, a)
|
exact <a>Ideal.IsHomogeneous.iInf₂</a> h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Nat.card_uIcc
|
a b c : ℕ
⊢ a ⊔ b + 1 - a ⊓ b = (↑b - ↑a).natAbs + 1
|
rw [← <a>Int.natCast_inj</a>, <a>sup_eq_max</a>, <a>inf_eq_min</a>, <a>Int.ofNat_sub</a>] <;> omega
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Nat.lean
|
List.rel_filter
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : (R ⇒ fun x x_1 => x ↔ x_1) (fun x => p x = true) fun x => q x = true
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
dsimp [<a>Relator.LiftFun</a>] at hpq
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Forall2.lean
|
List.rel_filter
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
by_cases h : p a
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : p a = true
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : ¬p a = true
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Forall2.lean
|
List.rel_filter
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : p a = true
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
have : q b := by rwa [← hpq h₁]
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : p a = true
this : q b = true
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Forall2.lean
|
List.rel_filter
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : p a = true
this : q b = true
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
simp only [<a>List.filter_cons_of_pos</a> _ h, <a>List.filter_cons_of_pos</a> _ this, <a>List.forall₂_cons</a>, h₁, <a>true_and_iff</a>, rel_filter hpq h₂]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Forall2.lean
|
List.rel_filter
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : p a = true
⊢ q b = true
|
rwa [← hpq h₁]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Forall2.lean
|
List.rel_filter
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : ¬p a = true
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
have : ¬q b := by rwa [← hpq h₁]
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : ¬p a = true
this : ¬q b = true
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Forall2.lean
|
List.rel_filter
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : ¬p a = true
this : ¬q b = true
⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))
|
simp only [<a>List.filter_cons_of_neg</a> _ h, <a>List.filter_cons_of_neg</a> _ this, rel_filter hpq h₂]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Forall2.lean
|
List.rel_filter
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : ¬p a = true
⊢ ¬q b = true
|
rwa [← hpq h₁]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Forall2.lean
|
Vector.map₂_flip
|
α : Type u_2
n : ℕ
β : Type u_3
xs : Vector α n
ys : Vector β n
γ : Type u_1
f : α → β → γ
⊢ map₂ f xs ys = map₂ (flip f) ys xs
|
induction xs, ys using <a>Vector.inductionOn₂</a> <;> simp_all[<a>flip</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Vector/MapLemmas.lean
|
OrderIso.refl_trans
|
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝² : LE α
inst✝¹ : LE β
inst✝ : LE γ
e : α ≃o β
⊢ (refl α).trans e = e
|
ext x
|
case h.h
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝² : LE α
inst✝¹ : LE β
inst✝ : LE γ
e : α ≃o β
x : α
⊢ ((refl α).trans e) x = e x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Hom/Basic.lean
|
OrderIso.refl_trans
|
case h.h
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝² : LE α
inst✝¹ : LE β
inst✝ : LE γ
e : α ≃o β
x : α
⊢ ((refl α).trans e) x = e x
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Hom/Basic.lean
|
TrivSqZeroExt.norm_inl
|
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹³ : SeminormedCommRing S
inst✝¹² : SeminormedRing R
inst✝¹¹ : SeminormedAddCommGroup M
inst✝¹⁰ : Algebra S R
inst✝⁹ : Module S M
inst✝⁸ : Module R M
inst✝⁷ : Module Rᵐᵒᵖ M
inst✝⁶ : BoundedSMul S R
inst✝⁵ : BoundedSMul S M
inst✝⁴ : BoundedSMul R M
inst✝³ : BoundedSMul Rᵐᵒᵖ M
inst✝² : SMulCommClass R Rᵐᵒᵖ M
inst✝¹ : IsScalarTower S R M
inst✝ : IsScalarTower S Rᵐᵒᵖ M
r : R
⊢ ‖inl r‖ = ‖r‖
|
simp [<a>TrivSqZeroExt.norm_def</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean
|
Stream'.WSeq.mem_of_mem_tail
|
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
⊢ a ∈ s.tail → a ∈ s
|
intro h
|
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
h : a ∈ s.tail
⊢ a ∈ s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
h : a ∈ s.tail
⊢ a ∈ s
|
have := h
|
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
h this : a ∈ s.tail
⊢ a ∈ s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
h this : a ∈ s.tail
⊢ a ∈ s
|
cases' h with n e
|
case intro
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
this : a ∈ s.tail
n : ℕ
e : (fun b => some (some a) = b) ((↑s.tail).get n)
⊢ a ∈ s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
this : a ∈ s.tail
n : ℕ
e : (fun b => some (some a) = b) ((↑s.tail).get n)
⊢ a ∈ s
|
revert s
|
case intro
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
⊢ ∀ {s : WSeq α}, a ∈ s.tail → (fun b => some (some a) = b) ((↑s.tail).get n) → a ∈ s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
⊢ ∀ {s : WSeq α}, a ∈ s.tail → (fun b => some (some a) = b) ((↑s.tail).get n) → a ∈ s
|
simp only [<a>Stream'.get</a>]
|
case intro
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
⊢ ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
⊢ ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
|
induction' n with n IH <;> intro s <;> induction' s using <a>Stream'.WSeq.recOn</a> with x s s <;> simp <;> intro m e <;> injections
|
case intro.zero.h2
α : Type u
β : Type v
γ : Type w
a x : α
s : WSeq α
m : a ∈ s
e : some (some a) = ↑s 0
⊢ a = x ∨ a ∈ s
case intro.succ.h2
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
x : α
s : WSeq α
m : a ∈ s
e : some (some a) = ↑s (n + 1)
⊢ a = x ∨ a ∈ s
case intro.succ.h3
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
s : WSeq α
m : a ∈ s.tail
e : some (some a) = ↑s.tail.think (n + 1)
⊢ a ∈ s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro.zero.h2
α : Type u
β : Type v
γ : Type w
a x : α
s : WSeq α
m : a ∈ s
e : some (some a) = ↑s 0
⊢ a = x ∨ a ∈ s
|
exact <a>Or.inr</a> m
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro.succ.h2
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
x : α
s : WSeq α
m : a ∈ s
e : some (some a) = ↑s (n + 1)
⊢ a = x ∨ a ∈ s
|
exact <a>Or.inr</a> m
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro.succ.h3
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
s : WSeq α
m : a ∈ s.tail
e : some (some a) = ↑s.tail.think (n + 1)
⊢ a ∈ s
|
apply IH m
|
case intro.succ.h3
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
s : WSeq α
m : a ∈ s.tail
e : some (some a) = ↑s.tail.think (n + 1)
⊢ some (some a) = ↑s.tail n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro.succ.h3
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
s : WSeq α
m : a ∈ s.tail
e : some (some a) = ↑s.tail.think (n + 1)
⊢ some (some a) = ↑s.tail n
|
rw [e]
|
case intro.succ.h3
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
s : WSeq α
m : a ∈ s.tail
e : some (some a) = ↑s.tail.think (n + 1)
⊢ ↑s.tail.think (n + 1) = ↑s.tail n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro.succ.h3
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
s : WSeq α
m : a ∈ s.tail
e : some (some a) = ↑s.tail.think (n + 1)
⊢ ↑s.tail.think (n + 1) = ↑s.tail n
|
cases <a>Stream'.WSeq.tail</a> s
|
case intro.succ.h3.mk
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
s : WSeq α
m : a ∈ s.tail
e : some (some a) = ↑s.tail.think (n + 1)
val✝ : Stream' (Option (Option α))
property✝ : val✝.IsSeq
⊢ ↑(think ⟨val✝, property✝⟩) (n + 1) = ↑⟨val✝, property✝⟩ n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_tail
|
case intro.succ.h3.mk
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
IH : ∀ {s : WSeq α}, a ∈ s.tail → some (some a) = ↑s.tail n → a ∈ s
s : WSeq α
m : a ∈ s.tail
e : some (some a) = ↑s.tail.think (n + 1)
val✝ : Stream' (Option (Option α))
property✝ : val✝.IsSeq
⊢ ↑(think ⟨val✝, property✝⟩) (n + 1) = ↑⟨val✝, property✝⟩ n
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Rat.normalize_num_den
|
n : Int
d : Nat
z : d ≠ 0
n' : Int
d' : Nat
z' : d' ≠ 0
c : n'.natAbs.Coprime d'
h : normalize n d z = { num := n', den := d', den_nz := z', reduced := c }
⊢ ∃ m, m ≠ 0 ∧ n = n' * ↑m ∧ d = d' * m
|
have := <a>Rat.normalize_num_den'</a> n d z
|
n : Int
d : Nat
z : d ≠ 0
n' : Int
d' : Nat
z' : d' ≠ 0
c : n'.natAbs.Coprime d'
h : normalize n d z = { num := n', den := d', den_nz := z', reduced := c }
this : ∃ d_1, d_1 ≠ 0 ∧ n = (normalize n d z).num * ↑d_1 ∧ d = (normalize n d z).den * d_1
⊢ ∃ m, m ≠ 0 ∧ n = n' * ↑m ∧ d = d' * m
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
.lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean
|
Rat.normalize_num_den
|
n : Int
d : Nat
z : d ≠ 0
n' : Int
d' : Nat
z' : d' ≠ 0
c : n'.natAbs.Coprime d'
h : normalize n d z = { num := n', den := d', den_nz := z', reduced := c }
this : ∃ d_1, d_1 ≠ 0 ∧ n = (normalize n d z).num * ↑d_1 ∧ d = (normalize n d z).den * d_1
⊢ ∃ m, m ≠ 0 ∧ n = n' * ↑m ∧ d = d' * m
|
rwa [h] at this
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
.lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (s ×ˢ Iic x)
|
by_cases hρ_zero : (ν a).<a>MeasureTheory.Measure.restrict</a> s = 0
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (s ×ˢ Iic x)
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (s ×ˢ Iic x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (s ×ˢ Iic x)
|
have h_nonempty : <a>Nonempty</a> { r' : ℚ // x < ↑r' } := by obtain ⟨r, hrx⟩ := <a>exists_rat_gt</a> x exact ⟨⟨r, hrx⟩⟩
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (s ×ˢ Iic x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (s ×ˢ Iic x)
|
rw [h, <a>MeasureTheory.lintegral_iInf_directed_of_measurable</a> hρ_zero fun q : { r' : ℚ // x < ↑r' } ↦ ?_]
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a = (κ a) (s ×ˢ Iic x)
case neg.hf_int
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∀ (b : { r' // x < ↑r' }), ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a ≠ ⊤
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ Directed (fun x x_1 => x ≥ x_1) fun r b => ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
q : { r' // x < ↑r' }
⊢ Measurable fun b => ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a = (κ a) (s ×ˢ Iic x)
case neg.hf_int
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∀ (b : { r' // x < ↑r' }), ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a ≠ ⊤
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ Directed (fun x x_1 => x ≥ x_1) fun r b => ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
q : { r' // x < ↑r' }
⊢ Measurable fun b => ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q)
|
rotate_left
|
case neg.hf_int
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∀ (b : { r' // x < ↑r' }), ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a ≠ ⊤
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ Directed (fun x x_1 => x ≥ x_1) fun r b => ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
q : { r' // x < ↑r' }
⊢ Measurable fun b => ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q)
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a = (κ a) (s ×ˢ Iic x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a = (κ a) (s ×ˢ Iic x)
|
simp_rw [<a>ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat_rat</a> hf _ _ hs]
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, (κ a) (s ×ˢ Iic ↑↑b) = (κ a) (s ×ˢ Iic x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, (κ a) (s ×ˢ Iic ↑↑b) = (κ a) (s ×ˢ Iic x)
|
rw [← <a>MeasureTheory.measure_iInter_eq_iInf</a>]
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ (κ a) (⋂ i, s ×ˢ Iic ↑↑i) = (κ a) (s ×ˢ Iic x)
case neg.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∀ (i : { r' // x < ↑r' }), MeasurableSet (s ×ˢ Iic ↑↑i)
case neg.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ Directed (fun x x_1 => x ⊇ x_1) fun b => s ×ˢ Iic ↑↑b
case neg.hfin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∃ i, (κ a) (s ×ˢ Iic ↑↑i) ≠ ⊤
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (s ×ˢ Iic x)
|
rw [hρ_zero, <a>MeasureTheory.lintegral_zero_measure</a>]
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
⊢ 0 = (κ a) (s ×ˢ Iic x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
⊢ 0 = (κ a) (s ×ˢ Iic x)
|
have ⟨q, hq⟩ := <a>exists_rat_gt</a> x
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
⊢ 0 = (κ a) (s ×ˢ Iic x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
⊢ 0 = (κ a) (s ×ˢ Iic x)
|
suffices κ a (s ×ˢ <a>Set.Iic</a> (q : ℝ)) = 0 by symm refine <a>MeasureTheory.measure_mono_null</a> (fun p ↦ ?_) this simp only [<a>Set.mem_prod</a>, <a>Set.mem_Iic</a>, <a>and_imp</a>] exact fun h1 h2 ↦ ⟨h1, h2.trans hq.le⟩
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
⊢ (κ a) (s ×ˢ Iic ↑q) = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
⊢ (κ a) (s ×ˢ Iic ↑q) = 0
|
suffices (κ a (s ×ˢ <a>Set.Iic</a> (q : ℝ))).<a>ENNReal.toReal</a> = 0 by rw [<a>ENNReal.toReal_eq_zero_iff</a>] at this simpa [<a>MeasureTheory.measure_ne_top</a>] using this
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
⊢ ((κ a) (s ×ˢ Iic ↑q)).toReal = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
⊢ ((κ a) (s ×ˢ Iic ↑q)).toReal = 0
|
rw [← hf.setIntegral a hs q]
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
⊢ ∫ (b : β) in s, f (a, b) q ∂ν a = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
⊢ ∫ (b : β) in s, f (a, b) q ∂ν a = 0
|
simp [hρ_zero]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
⊢ 0 = (κ a) (s ×ˢ Iic x)
|
symm
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
⊢ (κ a) (s ×ˢ Iic x) = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
⊢ (κ a) (s ×ˢ Iic x) = 0
|
refine <a>MeasureTheory.measure_mono_null</a> (fun p ↦ ?_) this
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
p : β × ℝ
⊢ p ∈ s ×ˢ Iic x → p ∈ s ×ˢ Iic ↑q
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
p : β × ℝ
⊢ p ∈ s ×ˢ Iic x → p ∈ s ×ˢ Iic ↑q
|
simp only [<a>Set.mem_prod</a>, <a>Set.mem_Iic</a>, <a>and_imp</a>]
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
p : β × ℝ
⊢ p.1 ∈ s → p.2 ≤ x → p.1 ∈ s ∧ p.2 ≤ ↑q
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
p : β × ℝ
⊢ p.1 ∈ s → p.2 ≤ x → p.1 ∈ s ∧ p.2 ≤ ↑q
|
exact fun h1 h2 ↦ ⟨h1, h2.trans hq.le⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : ((κ a) (s ×ˢ Iic ↑q)).toReal = 0
⊢ (κ a) (s ×ˢ Iic ↑q) = 0
|
rw [<a>ENNReal.toReal_eq_zero_iff</a>] at this
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0 ∨ (κ a) (s ×ˢ Iic ↑q) = ⊤
⊢ (κ a) (s ×ˢ Iic ↑q) = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : (ν a).restrict s = 0
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0 ∨ (κ a) (s ×ˢ Iic ↑q) = ⊤
⊢ (κ a) (s ×ˢ Iic ↑q) = 0
|
simpa [<a>MeasureTheory.measure_ne_top</a>] using this
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
|
congr with b : 1
|
case e_f.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) =
⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) =
⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)
|
simp_rw [← <a>ProbabilityTheory.measure_stieltjesOfMeasurableRat_Iic</a>]
|
case e_f.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic x) = ⨅ r, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑r)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic x) = ⨅ r, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑r)
|
rw [← <a>MeasureTheory.measure_iInter_eq_iInf</a>]
|
case e_f.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic x) = (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (⋂ i, Iic ↑↑i)
case e_f.h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ ∀ (i : { r' // x < ↑r' }), MeasurableSet (Iic ↑↑i)
case e_f.h.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ Directed (fun x x_1 => x ⊇ x_1) fun r => Iic ↑↑r
case e_f.h.hfin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ⊤
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic x) = (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (⋂ i, Iic ↑↑i)
|
congr with y : 1
|
case e_f.h.h.e_6.h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
⊢ y ∈ Iic x ↔ y ∈ ⋂ i, Iic ↑↑i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.h.e_6.h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
⊢ y ∈ Iic x ↔ y ∈ ⋂ i, Iic ↑↑i
|
simp only [<a>Set.mem_Iic</a>, <a>Set.mem_iInter</a>, <a>Subtype.forall</a>]
|
case e_f.h.h.e_6.h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
⊢ y ≤ x ↔ ∀ (a : ℚ), x < ↑a → y ≤ ↑a
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.h.e_6.h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
⊢ y ≤ x ↔ ∀ (a : ℚ), x < ↑a → y ≤ ↑a
|
refine ⟨fun h a ha ↦ h.trans ?_, fun h ↦ ?_⟩
|
case e_f.h.h.e_6.h.h.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a✝ : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a✝).restrict s = 0
b : β
y : ℝ
h : y ≤ x
a : ℚ
ha : x < ↑a
⊢ x ≤ ↑a
case e_f.h.h.e_6.h.h.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
h : ∀ (a : ℚ), x < ↑a → y ≤ ↑a
⊢ y ≤ x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.h.e_6.h.h.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a✝ : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a✝).restrict s = 0
b : β
y : ℝ
h : y ≤ x
a : ℚ
ha : x < ↑a
⊢ x ≤ ↑a
|
exact mod_cast ha.le
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.h.e_6.h.h.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
h : ∀ (a : ℚ), x < ↑a → y ≤ ↑a
⊢ y ≤ x
|
refine <a>le_of_forall_lt_rat_imp_le</a> fun q hq ↦ h q ?_
|
case e_f.h.h.e_6.h.h.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
h : ∀ (a : ℚ), x < ↑a → y ≤ ↑a
q : ℚ
hq : x < ↑q
⊢ x < ↑q
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.