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 |
|---|---|---|---|---|---|---|
Polynomial.rootSet_prod | R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝³ : Field R
p q : R[X]
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
ι : Type u_1
f : ι → R[X]
s : Finset ι
h : s.prod f ≠ 0
⊢ (s.prod f).rootSet S = ⋃ i ∈ s, (f i).rootSet S | simp only [<a>Polynomial.rootSet</a>, <a>Polynomial.aroots</a>, ← <a>Finset.mem_coe</a>] | R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝³ : Field R
p q : R[X]
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
ι : Type u_1
f : ι → R[X]
s : Finset ι
h : s.prod f ≠ 0
⊢ ↑(map (algebraMap R S) (s.prod f)).roots.toFinset = ⋃ i ∈ ↑s, ↑(map (algebraMap R S) (f i)).roots.toFinset | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Polynomial/FieldDivision.lean |
Polynomial.rootSet_prod | R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝³ : Field R
p q : R[X]
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
ι : Type u_1
f : ι → R[X]
s : Finset ι
h : s.prod f ≠ 0
⊢ ↑(map (algebraMap R S) (s.prod f)).roots.toFinset = ⋃ i ∈ ↑s, ↑(map (algebraMap R S) (f i)).roots.toFinset | rw [<a>Polynomial.map_prod</a>, <a>Polynomial.roots_prod</a>, <a>Finset.bind_toFinset</a>, s.val_toFinset, <a>Finset.coe_biUnion</a>] | case a
R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝³ : Field R
p q : R[X]
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
ι : Type u_1
f : ι → R[X]
s : Finset ι
h : s.prod f ≠ 0
⊢ ∏ i ∈ s, map (algebraMap R S) (f i) ≠ 0 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Polynomial/FieldDivision.lean |
Polynomial.rootSet_prod | case a
R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝³ : Field R
p q : R[X]
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
ι : Type u_1
f : ι → R[X]
s : Finset ι
h : s.prod f ≠ 0
⊢ ∏ i ∈ s, map (algebraMap R S) (f i) ≠ 0 | rwa [← <a>Polynomial.map_prod</a>, <a>Ne</a>, <a>Polynomial.map_eq_zero</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Polynomial/FieldDivision.lean |
Real.toNNReal_mul_nnnorm | α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
x y : ℝ
hx : 0 ≤ x
⊢ x.toNNReal * ‖y‖₊ = ‖x * y‖₊ | ext | case a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
x y : ℝ
hx : 0 ≤ x
⊢ ↑(x.toNNReal * ‖y‖₊) = ↑‖x * y‖₊ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/Normed/Field/Basic.lean |
Real.toNNReal_mul_nnnorm | case a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
x y : ℝ
hx : 0 ≤ x
⊢ ↑(x.toNNReal * ‖y‖₊) = ↑‖x * y‖₊ | simp only [<a>NNReal.coe_mul</a>, <a>nnnorm_mul</a>, <a>coe_nnnorm</a>, <a>Real.toNNReal_of_nonneg</a>, <a>Real.norm_of_nonneg</a>, hx, <a>NNReal.coe_mk</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/Normed/Field/Basic.lean |
CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : CoreHomEquiv F G
X' X : C
Y Y' : D
f : X ⟶ G.obj Y
g : Y ⟶ Y'
⊢ (adj.homEquiv X Y').symm (f ≫ G.map g) = (adj.homEquiv X Y).symm f ≫ g | rw [<a>Equiv.symm_apply_eq</a>] | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : CoreHomEquiv F G
X' X : C
Y Y' : D
f : X ⟶ G.obj Y
g : Y ⟶ Y'
⊢ f ≫ G.map g = (adj.homEquiv X Y') ((adj.homEquiv X Y).symm f ≫ g) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Adjunction/Basic.lean |
CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : CoreHomEquiv F G
X' X : C
Y Y' : D
f : X ⟶ G.obj Y
g : Y ⟶ Y'
⊢ f ≫ G.map g = (adj.homEquiv X Y') ((adj.homEquiv X Y).symm f ≫ g) | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Adjunction/Basic.lean |
Rat.toNNRat_inv | p q✝ q : ℚ
⊢ q⁻¹.toNNRat = q.toNNRat⁻¹ | obtain hq | hq := <a>le_total</a> q 0 | case inl
p q✝ q : ℚ
hq : q ≤ 0
⊢ q⁻¹.toNNRat = q.toNNRat⁻¹
case inr
p q✝ q : ℚ
hq : 0 ≤ q
⊢ q⁻¹.toNNRat = q.toNNRat⁻¹ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/NNRat/Lemmas.lean |
Rat.toNNRat_inv | case inl
p q✝ q : ℚ
hq : q ≤ 0
⊢ q⁻¹.toNNRat = q.toNNRat⁻¹ | rw [toNNRat_eq_zero.mpr hq, <a>GroupWithZero.inv_zero</a>, toNNRat_eq_zero.mpr (inv_nonpos.mpr hq)] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/NNRat/Lemmas.lean |
Rat.toNNRat_inv | case inr
p q✝ q : ℚ
hq : 0 ≤ q
⊢ q⁻¹.toNNRat = q.toNNRat⁻¹ | nth_rw 1 [← <a>Rat.coe_toNNRat</a> q hq] | case inr
p q✝ q : ℚ
hq : 0 ≤ q
⊢ (↑q.toNNRat)⁻¹.toNNRat = q.toNNRat⁻¹ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/NNRat/Lemmas.lean |
Rat.toNNRat_inv | case inr
p q✝ q : ℚ
hq : 0 ≤ q
⊢ (↑q.toNNRat)⁻¹.toNNRat = q.toNNRat⁻¹ | rw [← <a>NNRat.coe_inv</a>, <a>NNRat.toNNRat_coe</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/NNRat/Lemmas.lean |
Cardinal.mk_equiv_eq_arrow_of_lift_eq | α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
⊢ #(α ≃ β') = #(α → β') | obtain ⟨e⟩ := lift_mk_eq'.mp leq | case intro
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
e : α ≃ β'
⊢ #(α ≃ β') = #(α → β') | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/SetTheory/Cardinal/Ordinal.lean |
Cardinal.mk_equiv_eq_arrow_of_lift_eq | case intro
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
e : α ≃ β'
⊢ #(α ≃ β') = #(α → β') | have e₁ := lift_mk_eq'.mpr ⟨.equivCongr (.refl α) e⟩ | case intro
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
e : α ≃ β'
e₁ : lift.{max u v, u} #(α ≃ α) = lift.{u, max u v} #(α ≃ β')
⊢ #(α ≃ β') = #(α → β') | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/SetTheory/Cardinal/Ordinal.lean |
Cardinal.mk_equiv_eq_arrow_of_lift_eq | case intro
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
e : α ≃ β'
e₁ : lift.{max u v, u} #(α ≃ α) = lift.{u, max u v} #(α ≃ β')
⊢ #(α ≃ β') = #(α → β') | have e₂ := lift_mk_eq'.mpr ⟨.arrowCongr (.refl α) e⟩ | case intro
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
e : α ≃ β'
e₁ : lift.{max u v, u} #(α ≃ α) = lift.{u, max u v} #(α ≃ β')
e₂ : lift.{max u v, u} #(α → α) = lift.{u, max u v} #(α → β')
⊢ #(α ≃ β') = #(α → β') | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/SetTheory/Cardinal/Ordinal.lean |
Cardinal.mk_equiv_eq_arrow_of_lift_eq | case intro
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
e : α ≃ β'
e₁ : lift.{max u v, u} #(α ≃ α) = lift.{u, max u v} #(α ≃ β')
e₂ : lift.{max u v, u} #(α → α) = lift.{u, max u v} #(α → β')
⊢ #(α ≃ β') = #(α → β') | rw [<a>Cardinal.lift_id'</a>.{u,v}] at e₁ e₂ | case intro
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
e : α ≃ β'
e₁ : lift.{max u v, u} #(α ≃ α) = #(α ≃ β')
e₂ : lift.{max u v, u} #(α → α) = #(α → β')
⊢ #(α ≃ β') = #(α → β') | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/SetTheory/Cardinal/Ordinal.lean |
Cardinal.mk_equiv_eq_arrow_of_lift_eq | case intro
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
e : α ≃ β'
e₁ : lift.{max u v, u} #(α ≃ α) = #(α ≃ β')
e₂ : lift.{max u v, u} #(α → α) = #(α → β')
⊢ #(α ≃ β') = #(α → β') | rw [← e₁, ← e₂, <a>Cardinal.lift_inj</a>, <a>Cardinal.mk_perm_eq_self_power</a>, <a>Cardinal.power_def</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/SetTheory/Cardinal/Ordinal.lean |
Prod.comul_comp_inl | R : Type u
A : Type v
B : Type w
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : AddCommMonoid B
inst✝³ : Module R A
inst✝² : Module R B
inst✝¹ : Coalgebra R A
inst✝ : Coalgebra R B
⊢ comul ∘ₗ inl R A B = TensorProduct.map (inl R A B) (inl R A B) ∘ₗ comul | ext | case h
R : Type u
A : Type v
B : Type w
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : AddCommMonoid B
inst✝³ : Module R A
inst✝² : Module R B
inst✝¹ : Coalgebra R A
inst✝ : Coalgebra R B
x✝ : A
⊢ (comul ∘ₗ inl R A B) x✝ = (TensorProduct.map (inl R A B) (inl R A B) ∘ₗ comul) x✝ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Coalgebra/Basic.lean |
Prod.comul_comp_inl | case h
R : Type u
A : Type v
B : Type w
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : AddCommMonoid B
inst✝³ : Module R A
inst✝² : Module R B
inst✝¹ : Coalgebra R A
inst✝ : Coalgebra R B
x✝ : A
⊢ (comul ∘ₗ inl R A B) x✝ = (TensorProduct.map (inl R A B) (inl R A B) ∘ₗ comul) x✝ | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Coalgebra/Basic.lean |
Path.trans_pi_eq_pi_trans | X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ₀ : (i : ι) → Path (as i) (bs i)
γ₁ : (i : ι) → Path (bs i) (cs i)
⊢ (Path.pi γ₀).trans (Path.pi γ₁) = Path.pi fun i =... | ext t i | case a.h.h
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ₀ : (i : ι) → Path (as i) (bs i)
γ₁ : (i : ι) → Path (bs i) (cs i)
t : ↑I
i : ι
⊢ ((Path.pi γ₀).trans (Path... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Connected/PathConnected.lean |
Path.trans_pi_eq_pi_trans | case a.h.h
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ₀ : (i : ι) → Path (as i) (bs i)
γ₁ : (i : ι) → Path (bs i) (cs i)
t : ↑I
i : ι
⊢ ((Path.pi γ₀).trans (Path... | unfold <a>Path.trans</a> | case a.h.h
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ₀ : (i : ι) → Path (as i) (bs i)
γ₁ : (i : ι) → Path (bs i) (cs i)
t : ↑I
i : ι
⊢ {
toFun :=
... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Connected/PathConnected.lean |
Path.trans_pi_eq_pi_trans | case a.h.h
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ₀ : (i : ι) → Path (as i) (bs i)
γ₁ : (i : ι) → Path (bs i) (cs i)
t : ↑I
i : ι
⊢ {
toFun :=
... | simp only [<a>Path.coe_mk_mk</a>, <a>Function.comp_apply</a>, <a>Path.pi_coe</a>] | case a.h.h
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ₀ : (i : ι) → Path (as i) (bs i)
γ₁ : (i : ι) → Path (bs i) (cs i)
t : ↑I
i : ι
⊢ (if ↑t ≤ 1 / 2 then (Path... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Connected/PathConnected.lean |
Path.trans_pi_eq_pi_trans | case a.h.h
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ₀ : (i : ι) → Path (as i) (bs i)
γ₁ : (i : ι) → Path (bs i) (cs i)
t : ↑I
i : ι
⊢ (if ↑t ≤ 1 / 2 then (Path... | split_ifs <;> rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Connected/PathConnected.lean |
MvQPF.Cofix.abs_repr | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Cofix F α
⊢ Quot.mk Mcongr x.repr = x | let R := fun x y : <a>MvQPF.Cofix</a> F α => <a>MvQPF.Cofix.abs</a> (<a>MvQPF.Cofix.repr</a> y) = x | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Cofix F α
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
⊢ Quot.mk Mcongr x.repr = x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Cofix F α
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
⊢ Quot.mk Mcongr x.repr = x | refine <a>MvQPF.Cofix.bisim₂</a> R ?_ _ _ <a>rfl</a> | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Cofix F α
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
⊢ ∀ (x y : Cofix F α), R x y → LiftR' (α.RelLast' R) x.dest y.dest | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Cofix F α
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
⊢ ∀ (x y : Cofix F α), R x y → LiftR' (α.RelLast' R) x.dest y.dest | clear x | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
⊢ ∀ (x y : Cofix F α), R x y → LiftR' (α.RelLast' R) x.dest y.dest | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
⊢ ∀ (x y : Cofix F α), R x y → LiftR' (α.RelLast' R) x.dest y.dest | rintro x y h | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
x y : Cofix F α
h : R x y
⊢ LiftR' (α.RelLast' R) x.dest y.dest | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
x y : Cofix F α
h : R x y
⊢ LiftR' (α.RelLast' R) x.dest y.dest | subst h | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
y : Cofix F α
⊢ LiftR' (α.RelLast' R) (abs y.repr).dest y.dest | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
y : Cofix F α
⊢ LiftR' (α.RelLast' R) (abs y.repr).dest y.dest | dsimp [<a>MvQPF.Cofix.dest</a>, <a>MvQPF.Cofix.abs</a>] | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
y : Cofix F α
⊢ LiftR' (α.RelLast' R) ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) y.repr))
(Quot.lift (fun x => (TypeVe... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
y : Cofix F α
⊢ LiftR' (α.RelLast' R) ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) y.repr))
(Quot.lift (fun x => (TypeVe... | induction y using <a>Quot.ind</a> | case mk
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R) ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr (Quot.mk Mcongr a✝))))
... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | case mk
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R) ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr (Quot.mk Mcongr a✝))))
... | simp only [<a>MvQPF.Cofix.repr</a>, <a>MvPFunctor.M.dest_corec</a>, <a>MvQPF.abs_map</a>, <a>MvQPF.abs_repr</a>, <a>Function.comp</a>] | case mk
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R)
((TypeVec.id ::: Quot.mk Mcongr) <$$>
(TypeVec.id ::: M.corec (P F) fun x => MvQPF.r... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | case mk
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R)
((TypeVec.id ::: Quot.mk Mcongr) <$$>
(TypeVec.id ::: M.corec (P F) fun x => MvQPF.r... | conv => congr; rfl; rw [<a>MvQPF.Cofix.dest</a>] | case mk
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R)
((TypeVec.id ::: Quot.mk Mcongr) <$$>
(TypeVec.id :::
M.corec (P F) fun x ... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | case mk
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R)
((TypeVec.id ::: Quot.mk Mcongr) <$$>
(TypeVec.id :::
M.corec (P F) fun x ... | rw [<a>MvFunctor.map_map</a>, <a>MvFunctor.map_map</a>, ← <a>TypeVec.appendFun_comp_id</a>, ← <a>TypeVec.appendFun_comp_id</a>] | case mk
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R)
((TypeVec.id :::
(Quot.mk Mcongr ∘
M.corec (P F) fun x =>
... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | case mk
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R)
((TypeVec.id :::
(Quot.mk Mcongr ∘
M.corec (P F) fun x =>
... | apply <a>MvQPF.liftR_map_last</a> | case mk.hh
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ ∀ (x : (P F).M α),
R
(((Quot.mk Mcongr ∘
M.corec (P F) fun x =>
MvQPF.repr (Q... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | case mk.hh
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ ∀ (x : (P F).M α),
R
(((Quot.mk Mcongr ∘
M.corec (P F) fun x =>
MvQPF.repr (Q... | intros | case mk.hh
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ x✝ : (P F).M α
⊢ R
(((Quot.mk Mcongr ∘
M.corec (P F) fun x =>
MvQPF.repr (Quot.lift (fun x => (TypeVe... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
MvQPF.Cofix.abs_repr | case mk.hh
n : ℕ
F✝ : TypeVec.{u} (n + 1) → Type u
q✝ : MvQPF F✝
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ x✝ : (P F).M α
⊢ R
(((Quot.mk Mcongr ∘
M.corec (P F) fun x =>
MvQPF.repr (Quot.lift (fun x => (TypeVe... | rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean |
norm_deriv_le_of_lip' | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
x₀ : 𝕜
C : ℝ
hC₀ : 0 ≤ C
hlip : ∀ᶠ (x : 𝕜) in 𝓝... | simpa [<a>norm_deriv_eq_norm_fderiv</a>] using <a>norm_fderiv_le_of_lip'</a> 𝕜 hC₀ hlip | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/Calculus/Deriv/Basic.lean |
lt_one_div | ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
ha : 0 < a
hb : 0 < b
⊢ a < 1 / b ↔ b < 1 / a | simpa using <a>lt_inv</a> ha hb | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Order/Field/Basic.lean |
ProbabilityTheory.kernel.withDensity_apply | α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f : α → β → ℝ≥0∞
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
hf : Measurable (Function.uncurry f)
a : α
⊢ (withDensity κ f) a = (κ a).withDensity (f a) | classical rw [<a>ProbabilityTheory.kernel.withDensity</a>, <a>dif_pos</a> hf] rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Probability/Kernel/WithDensity.lean |
ProbabilityTheory.kernel.withDensity_apply | α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f : α → β → ℝ≥0∞
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
hf : Measurable (Function.uncurry f)
a : α
⊢ (withDensity κ f) a = (κ a).withDensity (f a) | rw [<a>ProbabilityTheory.kernel.withDensity</a>, <a>dif_pos</a> hf] | α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f : α → β → ℝ≥0∞
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
hf : Measurable (Function.uncurry f)
a : α
⊢ ⟨fun a => (κ a).withDensity (f a), ⋯⟩ a = (κ a).withDensity (f a) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Probability/Kernel/WithDensity.lean |
ProbabilityTheory.kernel.withDensity_apply | α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f : α → β → ℝ≥0∞
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
hf : Measurable (Function.uncurry f)
a : α
⊢ ⟨fun a => (κ a).withDensity (f a), ⋯⟩ a = (κ a).withDensity (f a) | rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Probability/Kernel/WithDensity.lean |
MeasureTheory.AEEqFun.compQuasiMeasurePreserving_eq_mk | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : MeasurableSpace α
μ ν✝ : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : MeasurableSpace β
ν : Measure β
f : α → β
g : β →ₘ[ν] γ
hf : QuasiMeasurePreserving f μ ν
⊢ g.compQuasiMeasurePreserving f hf = mk (... | rw [← <a>MeasureTheory.AEEqFun.compQuasiMeasurePreserving_mk</a> g.aestronglyMeasurable hf, <a>MeasureTheory.AEEqFun.mk_coeFn</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Function/AEEqFun.lean |
MeasureTheory.Measure.outerMeasure_le_iff | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
inst✝ : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
m : OuterMeasure α
⊢ m ≤ μ.toOuterMeasure ↔ ∀ (s : Set α), MeasurableSet s → m s ≤ μ s | simpa only [μ.trimmed] using <a>MeasureTheory.OuterMeasure.le_trim_iff</a> (m₂ := μ.1) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean |
CategoryTheory.Limits.kernelSubobjectMap_arrow | C : Type u
inst✝³ : Category.{v, u} C
X Y Z : C
inst✝² : HasZeroMorphisms C
f : X ⟶ Y
inst✝¹ : HasKernel f
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasKernel f'
sq : Arrow.mk f ⟶ Arrow.mk f'
⊢ kernelSubobjectMap sq ≫ (kernelSubobject f').arrow = (kernelSubobject f).arrow ≫ sq.left | simp [<a>CategoryTheory.Limits.kernelSubobjectMap</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Subobject/Limits.lean |
Batteries.RBNode.Stream.foldl_eq_foldl_toList | α : Type u_1
α✝ : Type u_2
f : α✝ → α → α✝
init : α✝
t : RBNode.Stream α
⊢ foldl f init t = List.foldl f init t.toList | induction t generalizing init <;> simp [*, <a>Batteries.RBNode.foldl_eq_foldl_toList</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean |
CategoryTheory.Cat.associator_hom_app | B C D E : Cat
F : B ⟶ C
G : C ⟶ D
H : D ⟶ E
X : ↑B
⊢ ((F ≫ G) ≫ H).obj X = (F ≫ G ≫ H).obj X | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Category/Cat.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0 | have hi₁ : <a>MeasurableSet</a> i := <a>by_contradiction</a> fun h => <a>ne_of_lt</a> hi <| s.not_measurable h | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0 | by_cases h : s ≤[i] 0 | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j... | by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq</a> s i l] 0 | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | swap | case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
¬∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n)... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | set A := i \ ⋃ l, <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq</a> s i l with hA | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | set bdd : ℕ → ℕ := fun n => <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.findExistsOneDivLT</a> s (i \ ⋃ k ≤ n, <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq</a> s i k) | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | have h₂ : s A ≤ s i := by rw [h₁] apply <a>le_add_of_nonneg_right</a> exact <a>tsum_nonneg</a> fun n => <a>le_of_lt</a> (<a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq</a> h _ (hn n)) | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | have h₃ : <a>Filter.Tendsto</a> (fun n => (bdd n : ℝ) + 1) <a>Filter.atTop</a> <a>Filter.atTop</a> := by simp only [<a>one_div</a>] at h₃' exact <a>Summable.tendsto_atTop_of_pos</a> h₃' fun n => <a>Nat.cast_add_one_pos</a> (bdd n) | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | have h₄ : <a>Filter.Tendsto</a> (fun n => (bdd n : ℝ)) <a>Filter.atTop</a> <a>Filter.atTop</a> := by convert atTop.tendsto_atTop_add_const_right (-1) h₃; simp | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | have A_meas : <a>MeasurableSet</a> A := hi₁.diff (<a>MeasurableSet.iUnion</a> fun _ => <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet</a> _) | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | refine ⟨A, A_meas, <a>Set.diff_subset</a>, ?_, h₂.trans_lt hi⟩ | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | by_contra hnn | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | rw [<a>MeasureTheory.VectorMeasure.restrict_le_restrict_iff</a> _ _ A_meas] at hnn | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | push_neg at hnn | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), M... | obtain ⟨E, hE₁, hE₂, hE₃⟩ := hnn | case pos.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l... | obtain ⟨k, hk₁, hk₂⟩ := this | case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restric... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restric... | have hA' : A ⊆ i \ ⋃ l ≤ k, <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq</a> s i l := by apply <a>Set.diff_subset_diff_right</a> intro x; simp only [<a>Set.mem_iUnion</a>] rintro ⟨n, _, hn₂⟩ exact ⟨n, hn₂⟩ | case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restric... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restric... | refine <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.findExistsOneDivLT_min</a> (hn' k) (<a>Nat.sub_lt</a> hk₁ <a>Nat.zero_lt_one</a>) ⟨E, <a>Set.Subset.trans</a> hE₂ hA', hE₁, ?_⟩ | case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restric... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restric... | convert hk₂ | case h.e'_3.h.e'_6
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ :... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h.e'_3.h.e'_6
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ :... | norm_cast | case h.e'_3.h.e'_6
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ :... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h.e'_3.h.e'_6
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ :... | exact <a>tsub_add_cancel_of_le</a> hk₁ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ... | exact ⟨i, hi₁, <a>Set.Subset.refl</a> _, h, hi⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
¬∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n)... | exact <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos'</a> hi₁ hi hn | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | intro n | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restri... | ext l | case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬re... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬re... | simp only [<a>exists_prop</a>, <a>Set.mem_iUnion</a>, <a>and_congr_left_iff</a>] | case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬re... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬re... | exact fun _ => Nat.lt_succ_iff.symm | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | rw [hA, ← s.of_disjoint_iUnion_nat, <a>add_comm</a>, <a>MeasureTheory.VectorMeasure.of_add_of_diff</a>] | case hA
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Me... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case hB
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Me... | exacts [hi₁, <a>Set.iUnion_subset</a> fun _ => <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_subset</a> _, fun _ => <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet</a> _, <a>_private.Mat... | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case hA
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Me... | exact <a>MeasurableSet.iUnion</a> fun _ => <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet</a> _ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | rw [h₁] | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | apply <a>le_add_of_nonneg_right</a> | case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Mea... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Mea... | exact <a>tsum_nonneg</a> fun n => <a>le_of_lt</a> (<a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq</a> h _ (hn n)) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | have : <a>Summable</a> fun l => s (<a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq</a> s i l) := <a>HasSum.summable</a> (s.m_iUnion (fun _ => <a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSe... | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | refine .of_nonneg_of_le (fun n => ?_) (fun n => ?_) (this.comp_injective <a>Nat.succ_injective</a>) | case refine_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < ... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case refine_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < ... | exact <a>le_of_lt</a> <a>Nat.one_div_pos_of_nat</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case refine_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < ... | exact <a>le_of_lt</a> (<a>_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_lt</a> n (hn' n)) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | simp only [<a>one_div</a>] at h₃' | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | exact <a>Summable.tendsto_atTop_of_pos</a> h₃' fun n => <a>Nat.cast_add_one_pos</a> (bdd n) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | convert atTop.tendsto_atTop_add_const_right (-1) h₃ | case h.e'_3.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < ... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h.e'_3.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < ... | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | rw [<a>Filter.tendsto_atTop_atTop</a>] at h₄ | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | obtain ⟨k, hk⟩ := h₄ (<a>Max.max</a> (1 / s E + 1) 1) | case intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n),... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n),... | refine ⟨k, ?_, ?_⟩ | case intro.refine_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case intro.refine_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | have hle := <a>le_of_max_le_right</a> (hk k <a>le_rfl</a>) | case intro.refine_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case intro.refine_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | norm_cast at hle | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case intro.refine_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | have : 1 / s E < bdd k := by linarith only [<a>le_of_max_le_left</a> (hk k <a>le_rfl</a>)] | case intro.refine_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case intro.refine_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | rw [<a>one_div</a>] at this ⊢ | case intro.refine_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case intro.refine_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ ... | rwa [<a>inv_lt</a> (<a>lt_trans</a> (<a>inv_pos</a>.2 hE₃) this) hE₃] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | linarith only [<a>le_of_max_le_left</a> (hk k <a>le_rfl</a>)] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureThe... | apply <a>Set.diff_subset_diff_right</a> | case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Mea... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Mea... | intro x | case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Mea... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Mea... | simp only [<a>Set.mem_iUnion</a>] | case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Mea... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), Mea... | rintro ⟨n, _, hn₂⟩ | case h.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ :... | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | case h.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
∀ (n : ℕ),
¬restrict s (i \ ⋃ l, ⋃ (_ :... | exact ⟨n, hn₂⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean |
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