mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.top.inl.refine_1
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ Ioi ↑x₁ ⊆ Ioi ↑a

pp: case a.inl.coe.top.inl.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑x₁)

Tactic:
·

Goal state after:
case a.inl.coe.top.inl.refine_1
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ Ioi ↑x₁ ⊆ Ioi ↑a
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case compl.refine_1
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
a : Set (Set α)
ha : a ∈ {x | ∃ f, {x | ∃ a, (↑(f a))ᶜ = x} = x}
t : Set α
ht : t ∈ a
⊢ t ∈ 𝒜 ∨ tᶜ ∈ 𝒜

pp: case compl.refine_2
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
⊢ uᶜ = ⋃ a ∈ {x | ∃ f, {x | ∃ a, (↑(f a))ᶜ = x} = x}, ⋂ t ∈ a, t

Tactic:
·

Goal state after:
case compl.refine_1
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
a : Set (Set α)
ha : a ∈ {x | ∃ f, {x | ∃ a, (↑(f a))ᶜ = x} = x}
t : Set α
ht : t ∈ a
⊢ t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.top.inl.refine_1
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ Ioi ↑x₁ ⊆ Ioi ↑a

pp: case a.inl.coe.top.inl.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑x₁)

Tactic:
· have : Ioi x₁ = ∅ := by
              have h : ∃ x, Ioi x = ∅ := ⟨a, ha⟩
              simp only [x₁, h, ↓reduceDIte]
              exact h.choose_spec
            simp [WithTop.Ioi_coe, this]

Goal state after:
case a.inl.coe.top.inl.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑x₁)
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: case inl
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hc0 : c = 0
⊢ False

pp: case inr
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hc0 : c ≠ 0
⊢ False

Tactic:
·

Goal state after:
case inl
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hc0 : c = 0
⊢ False
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.top.inl.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑x₁)

Tactic:
·

Goal state after:
case a.inl.coe.top.inl.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑x₁)
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.top.inl.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ Ioi ↑x₁ ∈ {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}

Tactic:
simp [d]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ m (⋃ i, f i) = m (⋃ i ∈ Finset.univ, f i)

pp: case convert_1
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ∀ i ∈ Finset.univ, f i ∈ C

pp: case convert_2
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ (↑Finset.univ).PairwiseDisjoint f

pp: case convert_3
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ⋃ i ∈ Finset.univ, f i ∈ C

Tactic:
·

Goal state after:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ m (⋃ i, f i) = m (⋃ i ∈ Finset.univ, f i)
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case compl.refine_2
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
⊢ uᶜ = ⋃ a ∈ {x | ∃ f, {x | ∃ a, (↑(f a))ᶜ = x} = x}, ⋂ t ∈ a, t

Tactic:
·

Goal state after:
case compl.refine_2
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
⊢ uᶜ = ⋃ a ∈ {x | ∃ f, {x | ∃ a, (↑(f a))ᶜ = x} = x}, ⋂ t ∈ a, t
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case convert_1
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ∀ i ∈ Finset.univ, f i ∈ C

pp: case convert_2
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ (↑Finset.univ).PairwiseDisjoint f

pp: case convert_3
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ⋃ i ∈ Finset.univ, f i ∈ C

Tactic:
·

Goal state after:
case convert_1
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ∀ i ∈ Finset.univ, f i ∈ C
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.top.inr
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
⊢ ∃ t ⊆ Ioi ↑a, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] t ∧ ⊤ ∈ t

Tactic:
·

Goal state after:
case a.inl.coe.top.inr
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
⊢ ∃ t ⊆ Ioi ↑a, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] t ∧ ⊤ ∈ t
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case convert_2
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ (↑Finset.univ).PairwiseDisjoint f

pp: case convert_3
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ⋃ i ∈ Finset.univ, f i ∈ C

Tactic:
·

Goal state after:
case convert_2
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ (↑Finset.univ).PairwiseDisjoint f
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case convert_3
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ⋃ i ∈ Finset.univ, f i ∈ C

Tactic:
·

Goal state after:
case convert_3
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ⋃ i ∈ Finset.univ, f i ∈ C
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.top.inr.refine_1
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb✝ : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
b : ι
bc' : b ∈ c'
hb : b ∈ Ioi a
⊢ Ioi ↑b ⊆ Ioi ↑a

pp: case a.inl.coe.top.inr.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb✝ : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
b : ι
bc' : b ∈ c'
hb : b ∈ Ioi a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑b)

Tactic:
·

Goal state after:
case a.inl.coe.top.inr.refine_1
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb✝ : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
b : ι
bc' : b ∈ c'
hb : b ∈ Ioi a
⊢ Ioi ↑b ⊆ Ioi ↑a
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case e'_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
⊢ m s + m t = ∑ i, m (![s, t] i)

pp: case convert_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
i : Fin (Nat.succ 0).succ
⊢ ![s, t] i ∈ C

pp: case convert_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
i j : Fin (Nat.succ 0).succ
hij : i ≠ j
⊢ (Disjoint on ![s, t]) i j

pp: case convert_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
⊢ ⋃ i, ![s, t] i ∈ C

Tactic:
·

Goal state after:
case e'_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
⊢ m s + m t = ∑ i, m (![s, t] i)
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case compl.refine_2.mp
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
hx : ∀ i ∈ A, ∃ i_2 ∈ i, x ∉ i_2
⊢ ∃ i, ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A), (↑(i ⟨i_2, ⋯⟩))ᶜ = i_1 → x ∈ i_1

pp: case compl.refine_2.mpr
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
hx : ∃ i, ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A), (↑(i ⟨i_2, ⋯⟩))ᶜ = i_1 → x ∈ i_1
⊢ ∀ i ∈ A, ∃ i_2 ∈ i, x ∉ i_2

Tactic:
·

Goal state after:
case compl.refine_2.mp
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
hx : ∀ i ∈ A, ∃ i_2 ∈ i, x ∉ i_2
⊢ ∃ i, ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A), (↑(i ⟨i_2, ⋯⟩))ᶜ = i_1 → x ∈ i_1
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.top.inr.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb✝ : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
b : ι
bc' : b ∈ c'
hb : b ∈ Ioi a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑b)

Tactic:
·

Goal state after:
case a.inl.coe.top.inr.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb✝ : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
b : ι
bc' : b ∈ c'
hb : b ∈ Ioi a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑b)
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case convert_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
i : Fin (Nat.succ 0).succ
⊢ ![s, t] i ∈ C

pp: case convert_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
i j : Fin (Nat.succ 0).succ
hij : i ≠ j
⊢ (Disjoint on ![s, t]) i j

pp: case convert_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
⊢ ⋃ i, ![s, t] i ∈ C

Tactic:
·

Goal state after:
case convert_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
i : Fin (Nat.succ 0).succ
⊢ ![s, t] i ∈ C
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case convert_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
i j : Fin (Nat.succ 0).succ
hij : i ≠ j
⊢ (Disjoint on ![s, t]) i j

pp: case convert_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
⊢ ⋃ i, ![s, t] i ∈ C

Tactic:
·

Goal state after:
case convert_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
i j : Fin (Nat.succ 0).succ
hij : i ≠ j
⊢ (Disjoint on ![s, t]) i j