Theorem: MeasureTheory.Measure.ext_of_Ico'
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean

tactic: rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, hsb, _⟩
state_before: α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
⊢ μ = ν

tactic: have : (⋃ (l ∈ s) (u ∈ s) (_ : l < u), {Ico l u} : Set (Set α)).Countable :=
  hsc.biUnion fun l _ => hsc.biUnion fun u _ => countable_iUnion fun _ => countable_singleton _
state_before: case intro.intro.intro.intro
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
⊢ μ = ν

tactic: simp only [← setOf_eq_eq_singleton, ← setOf_exists] at this
state_before: case intro.intro.intro.intro
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable (⋃ (l : α) (_ : l ∈ s) (u : α) (_ : u ∈ s) (_ : l < u), {Ico l u})
⊢ μ = ν

tactic: refine'
  Measure.ext_of_generateFrom_of_cover_subset
    (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico α)) (isPiSystem_Ico id id) _ this
    _ _ _
state_before: case intro.intro.intro.intro
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ μ = ν

tactic: rintro _ ⟨l, -, u, -, h, rfl⟩
state_before: case intro.intro.intro.intro.refine'_1
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ⊆ {S | ∃ l u, l < u ∧ Ico l u = S}

tactic: exact ⟨l, u, h, rfl⟩
state_before: case intro.intro.intro.intro.refine'_1.intro.intro.intro.intro.intro
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h✝ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
l u : α
h : l < u
⊢ Ico l u ∈ {S | ∃ l u, l < u ∧ Ico l u = S}

tactic: refine' sUnion_eq_univ_iff.2 fun x => _
state_before: case intro.intro.intro.intro.refine'_2
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ ⋃₀ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} = univ

tactic: rcases hsd.exists_le' hsb x with ⟨l, hls, hlx⟩
state_before: case intro.intro.intro.intro.refine'_2
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x✝ : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
x : α
⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b

tactic: rcases hsd.exists_gt x with ⟨u, hus, hxu⟩
state_before: case intro.intro.intro.intro.refine'_2.intro.intro
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x✝ : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
x l : α
hls : l ∈ s
hlx : l ≤ x
⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b

tactic: exact ⟨_, ⟨l, hls, u, hus, hlx.trans_lt hxu, rfl⟩, hlx, hxu⟩
state_before: case intro.intro.intro.intro.refine'_2.intro.intro.intro.intro
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x✝ : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
x l : α
hls : l ∈ s
hlx : l ≤ x
u : α
hus : u ∈ s
hxu : x < u
⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b

tactic: rintro _ ⟨l, -, u, -, hlt, rfl⟩
state_before: case intro.intro.intro.intro.refine'_3
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ ∀ (s_1 : Set α), s_1 ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} → ↑↑μ s_1 ≠ ⊤

tactic: exact hμ hlt
state_before: case intro.intro.intro.intro.refine'_3.intro.intro.intro.intro.intro
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
l u : α
hlt : l < u
⊢ ↑↑μ (Ico l u) ≠ ⊤

tactic: rintro _ ⟨l, u, hlt, rfl⟩
state_before: case intro.intro.intro.intro.refine'_4
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ ∀ (s : Set α), s ∈ {S | ∃ l u, l < u ∧ Ico l u = S} → ↑↑μ s = ↑↑ν s

tactic: exact h hlt
state_before: case intro.intro.intro.intro.refine'_4.intro.intro.intro
α✝ : Type ?u.682427
β : Type ?u.682430
γ : Type ?u.682433
γ₂ : Type ?u.682436
δ : Type ?u.682439
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.682532
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
l u : α
hlt : l < u
⊢ ↑↑μ (Ico l u) = ↑↑ν (Ico l u)