Theorem: MeasureTheory.lintegral_iInf_directed_of_measurable
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
state_after: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
val✝ : Encodable β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
tactic: cases nonempty_encodable β
state_before: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
val✝ : Encodable β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
state_after: case intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
val✝ : Encodable β
h✝ : IsEmpty β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ

case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
val✝ : Encodable β
h✝ : Nonempty β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
tactic: cases isEmpty_or_nonempty β
state_before: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
val✝ : Encodable β
h✝ : Nonempty β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
state_after: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
tactic: inhabit β
state_before: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
state_after: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ✝ ν : Measure α
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x x_1 => x ≥ x_1) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ (a : α), ⨅ b, f b a = ⨅ n, f (Directed.sequence f h_directed n) a
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
tactic: have : ∀ a, ⨅ b, f b a = ⨅ n, f (h_directed.sequence f n) a := by
    refine fun a =>
      le_antisymm (le_iInf fun n => iInf_le _ _)
        (le_iInf fun b => iInf_le_of_le (Encodable.encode b + 1) ?_)
    exact h_directed.sequence_le b a