mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case refine_4
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
h : ∑ u ∈ {s}, m u ≤ m t
⊢ m s ≤ m t

pp: case refine_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ↑{s} ⊆ C

pp: case refine_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ (↑{s}).PairwiseDisjoint id

pp: case refine_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ∀ s_1 ∈ {s}, s_1 ⊆ t

Tactic:
·

Goal state after:
case refine_4
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
h : ∑ u ∈ {s}, m u ≤ m t
⊢ m s ≤ m t
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case refine_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ↑{s} ⊆ C

pp: case refine_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ (↑{s}).PairwiseDisjoint id

pp: case refine_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ∀ s_1 ∈ {s}, s_1 ⊆ t

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ↑{s} ⊆ C
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case refine_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ (↑{s}).PairwiseDisjoint id

pp: case refine_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ∀ s_1 ∈ {s}, s_1 ⊆ t

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ (↑{s}).PairwiseDisjoint id
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case refine_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ∀ s_1 ∈ {s}, s_1 ⊆ t

Tactic:
·

Goal state after:
case refine_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ∀ s_1 ∈ {s}, s_1 ⊆ t
Module: Mathlib.MeasureTheory.Constructions.SimpleGraph

Goal state before:
pp: case mk
V : Type u_1
a✝ : Sym2 V
u v : V
⊢ Measurable fun x ↦ Quot.mk (Sym2.Rel V) (u, v) ∈ x.edgeSet

Tactic:
simp only [mem_edgeSet]

Goal state after:
case mk
V : Type u_1
a✝ : Sym2 V
u v : V
⊢ Measurable fun x ↦ x.Adj u v
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ ((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z) ∩ Iio ⊤ ⊆ Iio ↑a

pp: case a.inr.coe.coe.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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z) ∩ Iio ⊤)

Tactic:
·

Goal state after:
case a.inr.coe.coe.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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ ((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z) ∩ Iio ⊤ ⊆ Iio ↑a
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case refine_1
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ ↑(Finset.image f I) ⊆ C

pp: case refine_2
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ (↑(Finset.image f I)).PairwiseDisjoint id

pp: case refine_3
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ ∀ s ∈ Finset.image f I, s ⊆ ⋃ i, f i

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ ↑(Finset.image f I) ⊆ C
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case refine_2
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ (↑(Finset.image f I)).PairwiseDisjoint id

pp: case refine_3
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ ∀ s ∈ Finset.image f I, s ⊆ ⋃ i, f i

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ (↑(Finset.image f I)).PairwiseDisjoint id
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z) ∩ Iio ⊤)

Tactic:
·

Goal state after:
case a.inr.coe.coe.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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z) ∩ Iio ⊤)
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.refine_2.ht
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio ⊤)

pp: case a.inr.coe.coe.refine_2.hs
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] ((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z)

Tactic:
·

Goal state after:
case a.inr.coe.coe.refine_2.ht
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio ⊤)
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.refine_2.ht
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ Iio ⊤ ∈ {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}

Tactic:
simp [d]

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.refine_2.hs.hs
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (⋂ z ∈ f, Ioi ↑z)

pp: case a.inr.coe.coe.refine_2.hs.ht
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (⋂ z ∈ g, Iio ↑z)

Tactic:
·

Goal state after:
case a.inr.coe.coe.refine_2.hs.hs
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (⋂ z ∈ f, Ioi ↑z)
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case refine_3
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ ∀ s ∈ Finset.image f I, s ⊆ ⋃ i, f i

Tactic:
·

Goal state after:
case refine_3
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ ∀ s ∈ Finset.image f I, s ⊆ ⋃ i, f i
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.refine_2.hs.hs.hs
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ f.Finite

pp: case a.inr.coe.coe.refine_2.hs.hs.h
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ ∀ i ∈ f, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑i)

Tactic:
·

Goal state after:
case a.inr.coe.coe.refine_2.hs.hs.hs
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ f.Finite
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
u : α
h : u ≤ u
h' : ∃ p, p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2
u' : α := h'.choose.1
v' : α := h'.choose.2
hu'v' : u' ≤ v' ∧ Set.Ioc u u = Set.Ioc u' v'
⊢ f v' - f u' = f u - f u

pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
u v : α
h : u ≤ v
h' : ∃ p, p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2
u' : α := h'.choose.1
v' : α := h'.choose.2
hu'v' : u' ≤ v' ∧ Set.Ioc u v = Set.Ioc u' v'
huv : u < v
⊢ f v' - f u' = f v - f u

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
u : α
h : u ≤ u
h' : ∃ p, p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2
u' : α := h'.choose.1
v' : α := h'.choose.2
hu'v' : u' ≤ v' ∧ Set.Ioc u u = Set.Ioc u' v'
⊢ f v' - f u' = f u - f u
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.refine_2.hs.hs.h
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ ∀ i ∈ f, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑i)

Tactic:
·

Goal state after:
case a.inr.coe.coe.refine_2.hs.hs.h
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 b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ ∀ i ∈ f, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑i)
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case succ.inl
α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u : α
huv : u ≤ u
h'uv : ⋃₀ ↑I = Set.Ioc u u
⊢ onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u

pp: case succ.inr
α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
⊢ onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u

Tactic:
·

Goal state after:
case succ.inl
α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u : α
huv : u ≤ u
h'uv : ⋃₀ ↑I = Set.Ioc u u
⊢ onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u