-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
lemma inf_apply {s : Set α} (hs : MeasurableSet s) :
    (μ ⊓ ν) s = sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}

Proof state:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n => if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n => if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
x : α
hx : x ∈ s
hxt : x ∈ t
⊢ x ∈ iUnion t'

Tactic:
refine mem_iUnion.2 ⟨0, ?_⟩

Result:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n => if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n => if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
x : α
hx : x ∈ s
hxt : x ∈ t
⊢ x ∈ if 0 = 0 then t ∩ s else if 0 = 1 then tᶜ ∩ s else ∅

Proof state:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
⊢ μ (t ∩ s) + ν (tᶜ ∩ s) ≤ ∑' (n : ℕ), μ (t' n) ⊓ ν (t' n)

Tactic:
have hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n := by
  simp_rw [ht, compl_iUnion]
  refine fun x ⟨hx₁, hx₂⟩ ↦ mem_iUnion₂.2 ?_
  obtain ⟨i, hi⟩ := mem_iUnion.1 <| ht' hx₂
  refine ⟨i, ?_, hi⟩
  by_contra h
  simp only [mem_setOf_eq, not_lt] at h
  exact mem_iInter₂.1 hx₁ i h hi

Result:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
⊢ μ (t ∩ s) + ν (tᶜ ∩ s) ≤ ∑' (n : ℕ), μ (t' n) ⊓ ν (t' n)

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n => if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n => if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
⊢ μ (t' 0) ≤ μ (t ∩ s)

Tactic:
simp [ht']

Result:
no goals

Proof state:
case refine_2.refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
⊢ ∀ (a : ℕ) (b : a ∈ {k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑⟨a, b⟩) = μ (t' ↑⟨a, b⟩) ⊓ ν (t' ↑⟨a, b⟩)

Tactic:
intro n hn

Result:
case refine_2.refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
n : ℕ
hn : n ∈ {k | μ (t' k) ≤ ν (t' k)}
⊢ μ (t' ↑⟨n, hn⟩) = μ (t' ↑⟨n, hn⟩) ⊓ ν (t' ↑⟨n, hn⟩)

Proof state:
case intro
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
x : α
x✝ : x ∈ (⋂ i ∈ {k | μ (t' k) ≤ ν (t' k)}, (t' i)ᶜ) ∩ s
hx₁ : x ∈ ⋂ i ∈ {k | μ (t' k) ≤ ν (t' k)}, (t' i)ᶜ
hx₂ : x ∈ s
i : ℕ
hi : x ∈ t' i
h : i ∉ {k | ν (t' k) < μ (t' k)}
⊢ False

Tactic:
simp only [mem_setOf_eq, not_lt] at h

Result:
case intro
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
x : α
x✝ : x ∈ (⋂ i ∈ {k | μ (t' k) ≤ ν (t' k)}, (t' i)ᶜ) ∩ s
hx₁ : x ∈ ⋂ i ∈ {k | μ (t' k) ≤ ν (t' k)}, (t' i)ᶜ
hx₂ : x ∈ s
i : ℕ
hi : x ∈ t' i
h : μ (t' i) ≤ ν (t' i)
⊢ False

Proof state:
case intro
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
x : α
x✝ : x ∈ (⋂ i ∈ {k | μ (t' k) ≤ ν (t' k)}, (t' i)ᶜ) ∩ s
hx₁ : x ∈ ⋂ i ∈ {k | μ (t' k) ≤ ν (t' k)}, (t' i)ᶜ
hx₂ : x ∈ s
i : ℕ
hi : x ∈ t' i
h : μ (t' i) ≤ ν (t' i)
⊢ False

Tactic:
exact mem_iInter₂.1 hx₁ i h hi

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
⊢ {k | μ (t' k) ≤ ν (t' k)} ∩ {k | ν (t' k) < μ (t' k)} ⊆ ∅

Tactic:
rintro k ⟨hk₁, hk₂⟩

Result:
case intro
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
k : ℕ
hk₁ : k ∈ {k | μ (t' k) ≤ ν (t' k)}
hk₂ : k ∈ {k | ν (t' k) < μ (t' k)}
⊢ k ∈ ∅

Proof state:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
⊢ sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)} ≤ ∑' (n : ℕ), μ (t' n) ⊓ ν (t' n)

Tactic:
set t := ⋃ n ∈ {k : ℕ | μ (t' k) ≤ ν (t' k)}, t' n with ht

Result:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
⊢ sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)} ≤ ∑' (n : ℕ), μ (t' n) ⊓ ν (t' n)

Proof state:
case h
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
k : ℕ
⊢ k ∈ {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} ↔ k ∈ univ

Tactic:
simp [le_or_lt]

Result:
no goals