zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_106.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (ω' : Fin n → Ω × Ω) :
  |2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I| ≤ ↑n * (b + b) + C := sorry

Tactic state before:
case right.h
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n : ℕ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
ω' : Fin n → Ω × Ω
⊢ |2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I| ≤ ↑n * (b + b) + C

Tactic:
exact bound_σsum h𝓕' hC ω'

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_107.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n : ℕ} (ω : Ω × Ω)
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) :
  let μ2n := Measure.pi fun x => μ.prod μ;
  Integrable
    (fun a =>
      2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
    μ2n := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n : ℕ
ω : Ω × Ω
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ let μ2n := Measure.pi fun x => μ.prod μ;
  Integrable
    (fun a =>
      2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
    μ2n

Tactic:
exact integrable₁' h𝓕 h𝓕' (boundedness₀ h𝓕' hC ω)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_108.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    AEStronglyMeasurable
      (fun a =>
        ∫ (x : Fin n → Ω × Ω),
          (fun ω' =>
              2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X a.1) - f I (X a.2) + c I))
            x ∂Measure.pi fun x => μ.prod μ)
      (μ.prod μ)) :
  StronglyMeasurable fun a =>
    ∫ (x : Fin n → Ω × Ω),
      (fun ω' =>
          2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X a.1) - f I (X a.2) + c I))
        x ∂Measure.pi fun x => μ.prod μ := sorry

Tactic state before:
case left
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ AEStronglyMeasurable
    (fun a =>
      ∫ (x : Fin n → Ω × Ω),
        (fun ω' =>
            2⁻¹ ^ n *
              ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X a.1) - f I (X a.2) + c I))
          x ∂Measure.pi fun x => μ.prod μ)
    (μ.prod μ)

Tactic:
apply StronglyMeasurable.aestronglyMeasurable

Tactic state after:
[case left.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ StronglyMeasurable fun a =>
     ∫ (x : Fin n → Ω × Ω),
       (fun ω' =>
           2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X a.1) - f I (X a.2) + c I))
         x ∂Measure.pi fun x => μ.prod μ]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_109.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    StronglyMeasurable fun a =>
      ∫ (x : Fin n → Ω × Ω),
        (fun ω' =>
            2⁻¹ ^ n *
              ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X a.1) - f I (X a.2) + c I))
          x ∂Measure.pi fun x => μ.prod μ) :
  StronglyMeasurable fun ω =>
    2⁻¹ ^ n *
      ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I) := sorry

Tactic state before:
case left.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ StronglyMeasurable fun a =>
    ∫ (x : Fin n → Ω × Ω),
      (fun ω' =>
          2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X a.1) - f I (X a.2) + c I))
        x ∂Measure.pi fun x => μ.prod μ

Tactic:
apply
  @StronglyMeasurable.integral_prod_right' (Ω × Ω) (Fin n → (Ω × Ω)) ℝ _ _ (Measure.pi fun _ ↦ μ.prod μ) _ _ _
    (fun ω ↦
      (2⁻¹ ^ n *
        ∑ σ : Fin n → ({-1, 1} : Finset ℤ),
          ⨆ I, ∑ i : Fin n, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I)))

Tactic state after:
[case left.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ StronglyMeasurable fun ω =>
     2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_110.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    StronglyMeasurable fun ω =>
      2⁻¹ ^ n *
        ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I)) :
  Measurable fun ω =>
    2⁻¹ ^ n *
      ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I) := sorry

Tactic state before:
case left.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ StronglyMeasurable fun ω =>
    2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I)

Tactic:
apply Measurable.stronglyMeasurable

Tactic state after:
[case left.hf.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ Measurable fun ω =>
     2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_111.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    Measurable fun ω =>
      2⁻¹ ^ n *
        ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I)) :
  Measurable fun x =>
    ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x.2 i).1) - f I (X (x.2 i).2)) + (f I (X x.1.1) - f I (X x.1.2) + c I) := sorry

Tactic state before:
case left.hf.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ Measurable fun ω =>
    2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I)

Tactic:
apply Measurable.const_mul

Tactic state after:
[case left.hf.hf.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ Measurable fun x =>
     ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x.2 i).1) - f I (X (x.2 i).2)) + (f I (X x.1.1) - f I (X x.1.2) + c I)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_112.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    Measurable fun x =>
      ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x.2 i).1) - f I (X (x.2 i).2)) + (f I (X x.1.1) - f I (X x.1.2) + c I))
  (i : Fin n → { x // x ∈ {-1, 1} }) :
  i ∈ Finset.univ →
    Measurable fun a =>
      ⨆ I,
        ∑ i_1, ↑↑(i i_1) * (f I (X (a.2 i_1).1) - f I (X (a.2 i_1).2)) + (f I (X a.1.1) - f I (X a.1.2) + c I) := sorry

Tactic state before:
case left.hf.hf.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ Measurable fun x =>
    ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x.2 i).1) - f I (X (x.2 i).2)) + (f I (X x.1.1) - f I (X x.1.2) + c I)

Tactic:
apply Finset.measurable_sum Finset.univ

Tactic state after:
[case left.hf.hf.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ ∀ i ∈ Finset.univ,
     Measurable fun a =>
       ⨆ I, ∑ i_1, ↑↑(i i_1) * (f I (X (a.2 i_1).1) - f I (X (a.2 i_1).2)) + (f I (X a.1.1) - f I (X a.1.2) + c I)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_113.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (σ : Fin n → { x // x ∈ {-1, 1} }) (a : σ ∈ Finset.univ)
  (h_original_goal :
    Measurable fun a =>
      ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2)) + (f I (X a.1.1) - f I (X a.1.2) + c I))
  (i : ι) :
  Measurable fun b =>
    ∑ i_1, ↑↑(σ i_1) * (f i (X (b.2 i_1).1) - f i (X (b.2 i_1).2)) + (f i (X b.1.1) - f i (X b.1.2) + c i) := sorry

Tactic state before:
case left.hf.hf.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
σ : Fin n → { x // x ∈ {-1, 1} }
a✝ : σ ∈ Finset.univ
⊢ Measurable fun a =>
    ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2)) + (f I (X a.1.1) - f I (X a.1.2) + c I)

Tactic:
apply Measurable.iSup

Tactic state after:
[case left.hf.hf.hf.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 σ : Fin n → { x // x ∈ {-1, 1} }
 a✝ : σ ∈ Finset.univ
 ⊢ ∀ (i : ι),
     Measurable fun b =>
       ∑ i_1, ↑↑(σ i_1) * (f i (X (b.2 i_1).1) - f i (X (b.2 i_1).2)) + (f i (X b.1.1) - f i (X b.1.2) + c i)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_114.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (σ : Fin n → { x // x ∈ {-1, 1} }) (a : σ ∈ Finset.univ) (I : ι)
  (h_original_goal :
    Measurable fun b =>
      ∑ i, ↑↑(σ i) * (f I (X (b.2 i).1) - f I (X (b.2 i).2)) + (f I (X b.1.1) - f I (X b.1.2) + c I)) :
  Measurable fun a => ∑ i, ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2)) := sorry

Tactic state before:
case left.hf.hf.hf.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
σ : Fin n → { x // x ∈ {-1, 1} }
a✝ : σ ∈ Finset.univ
I : ι
⊢ Measurable fun b => ∑ i, ↑↑(σ i) * (f I (X (b.2 i).1) - f I (X (b.2 i).2)) + (f I (X b.1.1) - f I (X b.1.2) + c I)

Tactic:
apply Measurable.add

Tactic state after:
[case left.hf.hf.hf.hf.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 σ : Fin n → { x // x ∈ {-1, 1} }
 a✝ : σ ∈ Finset.univ
 I : ι
 ⊢ Measurable fun a => ∑ i, ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2)),
 case left.hf.hf.hf.hf.hg
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 σ : Fin n → { x // x ∈ {-1, 1} }
 a✝ : σ ∈ Finset.univ
 I : ι
 ⊢ Measurable fun a => f I (X a.1.1) - f I (X a.1.2) + c I]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_115.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (σ : Fin n → { x // x ∈ {-1, 1} }) (a : σ ∈ Finset.univ) (I : ι)
  (h_original_goal :
    Measurable fun b =>
      ∑ i, ↑↑(σ i) * (f I (X (b.2 i).1) - f I (X (b.2 i).2)) + (f I (X b.1.1) - f I (X b.1.2) + c I)) :
  Measurable fun a => f I (X a.1.1) - f I (X a.1.2) + c I := sorry

Tactic state before:
case left.hf.hf.hf.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
σ : Fin n → { x // x ∈ {-1, 1} }
a✝ : σ ∈ Finset.univ
I : ι
⊢ Measurable fun b => ∑ i, ↑↑(σ i) * (f I (X (b.2 i).1) - f I (X (b.2 i).2)) + (f I (X b.1.1) - f I (X b.1.2) + c I)

Tactic:
apply Measurable.add

Tactic state after:
[case left.hf.hf.hf.hf.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 σ : Fin n → { x // x ∈ {-1, 1} }
 a✝ : σ ∈ Finset.univ
 I : ι
 ⊢ Measurable fun a => ∑ i, ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2)),
 case left.hf.hf.hf.hf.hg
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 σ : Fin n → { x // x ∈ {-1, 1} }
 a✝ : σ ∈ Finset.univ
 I : ι
 ⊢ Measurable fun a => f I (X a.1.1) - f I (X a.1.2) + c I]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_116.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (σ : Fin n → { x // x ∈ {-1, 1} }) (a : σ ∈ Finset.univ) (I : ι)
  (h_original_goal : Measurable fun a => ∑ i, ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2))) (i : Fin n) :
  i ∈ Finset.univ → Measurable fun a => ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2)) := sorry

Tactic state before:
case left.hf.hf.hf.hf.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
σ : Fin n → { x // x ∈ {-1, 1} }
a✝ : σ ∈ Finset.univ
I : ι
⊢ Measurable fun a => ∑ i, ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2))

Tactic:
apply Finset.measurable_sum Finset.univ

Tactic state after:
[case left.hf.hf.hf.hf.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 σ : Fin n → { x // x ∈ {-1, 1} }
 a✝ : σ ∈ Finset.univ
 I : ι
 ⊢ ∀ i ∈ Finset.univ, Measurable fun a => ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2))]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_117.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (σ : Fin n → { x // x ∈ {-1, 1} }) (a : σ ∈ Finset.univ) (I : ι)
  (i : Fin n) (a_1 : i ∈ Finset.univ)
  (h_original_goal : Measurable fun a => ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2))) :
  Measurable fun x => f I (X (x.2 i).1) - f I (X (x.2 i).2) := sorry

Tactic state before:
case left.hf.hf.hf.hf.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
σ : Fin n → { x // x ∈ {-1, 1} }
a✝¹ : σ ∈ Finset.univ
I : ι
i : Fin n
a✝ : i ∈ Finset.univ
⊢ Measurable fun a => ↑↑(σ i) * (f I (X (a.2 i).1) - f I (X (a.2 i).2))

Tactic:
apply Measurable.const_mul

Tactic state after:
[case left.hf.hf.hf.hf.hf.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 σ : Fin n → { x // x ∈ {-1, 1} }
 a✝¹ : σ ∈ Finset.univ
 I : ι
 i : Fin n
 a✝ : i ∈ Finset.univ
 ⊢ Measurable fun x => f I (X (x.2 i).1) - f I (X (x.2 i).2)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_118.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (σ : Fin n → { x // x ∈ {-1, 1} }) (a : σ ∈ Finset.univ) (I : ι)
  (i : Fin n) (a_1 : i ∈ Finset.univ) : Measurable fun x => f I (X (x.2 i).1) - f I (X (x.2 i).2) := sorry

Tactic state before:
case left.hf.hf.hf.hf.hf.hf
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
σ : Fin n → { x // x ∈ {-1, 1} }
a✝¹ : σ ∈ Finset.univ
I : ι
i : Fin n
a✝ : i ∈ Finset.univ
⊢ Measurable fun x => f I (X (x.2 i).1) - f I (X (x.2 i).2)

Tactic:
exact (measurable_sub_part h𝓕).comp measurable_snd

Tactic state after: