zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_119.{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 => f I (X a.1.1) - f I (X a.1.2) + c I) :
  Measurable fun x => f I (X x.1.1) - f I (X x.1.2) := sorry

Tactic state before:
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

Tactic:
apply Measurable.add_const

Tactic state after:
[case left.hf.hf.hf.hf.hg.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 x => f I (X x.1.1) - f I (X x.1.2)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_120.{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 x => f I (X x.1.1) - f I (X x.1.2)) : Measurable fun a => f I (X a.1.1) := sorry

Tactic state before:
case left.hf.hf.hf.hf.hg.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 x => f I (X x.1.1) - f I (X x.1.2)

Tactic:
apply Measurable.sub

Tactic state after:
[case left.hf.hf.hf.hf.hg.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 => f I (X a.1.1),
 case left.hf.hf.hf.hf.hg.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.1.2)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_121.{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 x => f I (X x.1.1) - f I (X x.1.2)) : Measurable fun a => f I (X a.1.2) := sorry

Tactic state before:
case left.hf.hf.hf.hf.hg.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 x => f I (X x.1.1) - f I (X x.1.2)

Tactic:
apply Measurable.sub

Tactic state after:
[case left.hf.hf.hf.hf.hg.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 => f I (X a.1.1),
 case left.hf.hf.hf.hf.hg.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.1.2)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_122.{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) :
  Measurable fun a => f I (X a.1.1) := sorry

Tactic state before:
case left.hf.hf.hf.hf.hg.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 => f I (X a.1.1)

Tactic:
exact (h𝓕 I).comp <| measurable_fst.comp measurable_fst

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_123.{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) :
  Measurable fun a => f I (X a.1.2) := sorry

Tactic state before:
case left.hf.hf.hf.hf.hg.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.1.2)

Tactic:
exact (h𝓕 I).comp <| measurable_snd.comp measurable_fst

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_124.{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 :
    HasFiniteIntegral
      (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)) :
  ∀ᵐ (a : Ω × Ω) ∂μ.prod μ,
    ‖∫ (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‖ ≤
      ↑n * (b + b) + (b + b + C) := sorry

Tactic state before:
case right
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
⊢ HasFiniteIntegral
    (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 @MeasureTheory.hasFiniteIntegral_of_bounded _ _ _ _ _ _ _ (n * (b + b) + (b + b + C))

Tactic state after:
[case right
 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
 ⊢ ∀ᵐ (a : Ω × Ω) ∂μ.prod μ,
     ‖∫ (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‖ ≤
       ↑n * (b + b) + (b + b + C)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_125.{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) (ω' : Ω × Ω)
  (h_original_goal :
    |∫ (x : Fin n → Ω × Ω),
          2⁻¹ ^ n *
            ∑ σ,
              ⨆ I,
                ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) +
                  (f I (X ω'.1) - f I (X ω'.2) + c I) ∂Measure.pi fun x => μ.prod μ| ≤
      ↑n * (b + b) + (b + b + C)) :
  ∀ᵐ (x : Fin n → Ω × Ω) ∂Measure.pi fun x => μ.prod μ,
    |2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) + (f I (X ω'.1) - f I (X ω'.2) + c I)| ≤
      ↑n * (b + b) + (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
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
ω' : Ω × Ω
⊢ |∫ (x : Fin n → Ω × Ω),
        2⁻¹ ^ n *
          ∑ σ,
            ⨆ I,
              ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) +
                (f I (X ω'.1) - f I (X ω'.2) + c I) ∂Measure.pi fun x => μ.prod μ| ≤
    ↑n * (b + b) + (b + b + C)

Tactic:
apply abs_expectation_le_of_abs_le_const

Tactic state after:
[case right.h.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
 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
 ω' : Ω × Ω
 ⊢ ∀ᵐ (x : Fin n → Ω × Ω) ∂Measure.pi fun x => μ.prod μ,
     |2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) + (f I (X ω'.1) - f I (X ω'.2) + c I)| ≤
       ↑n * (b + b) + (b + b + C)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_126.{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) (ω' : Ω × Ω) (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)) + (f I (X ω'.1) - f I (X ω'.2) + c I)| ≤
    ↑n * (b + b) + (b + b + C) := sorry

Tactic state before:
case right.h.h.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
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
ω' : Ω × Ω
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)) + (f I (X ω'.1) - f I (X ω'.2) + c I)| ≤
    ↑n * (b + b) + (b + b + C)

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

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_127.{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 ω =>
        2⁻¹ ^ (n + 1) *
          ∑ σ,
            ⨆ I,
              ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
                (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I))
      ((μ.prod μ).prod (Measure.pi fun x => μ.prod μ))) :
  Measurable fun ω =>
    2⁻¹ ^ (n + 1) *
      ∑ σ,
        ⨆ I,
          ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
            (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I) := 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 ω =>
      2⁻¹ ^ (n + 1) *
        ∑ σ,
          ⨆ I,
            ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
              (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I))
    ((μ.prod μ).prod (Measure.pi fun x => μ.prod μ))

Tactic:
apply Measurable.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
 ⊢ Measurable fun ω =>
     2⁻¹ ^ (n + 1) *
       ∑ σ,
         ⨆ I,
           ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
             (↑↑(σ ↑n) * (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_128.{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 + 1) *
        ∑ σ,
          ⨆ I,
            ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
              (↑↑(σ ↑n) * (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)) +
          (↑↑(σ ↑n) * (f I (X x.1.1) - f I (X x.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
⊢ Measurable fun ω =>
    2⁻¹ ^ (n + 1) *
      ∑ σ,
        ⨆ I,
          ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (↑↑(σ ↑n) * (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
 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)) + (↑↑(σ ↑n) * (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_129.{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)) + (↑↑(σ ↑n) * (f I (X x.1.1) - f I (X x.1.2)) + c I))
  (i : Fin (n + 1) → { 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)) +
          (↑↑(i ↑n) * (f I (X a.1.1) - f I (X a.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 x =>
    ∑ σ,
      ⨆ I, ∑ i, ↑↑(σ ↑↑i) * (f I (X (x.2 i).1) - f I (X (x.2 i).2)) + (↑↑(σ ↑n) * (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
 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)) +
           (↑↑(i ↑n) * (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_130.{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 + 1) → { 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)) + (↑↑(σ ↑n) * (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)) +
      (↑↑(σ ↑n) * (f i (X b.1.1) - f i (X b.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
σ : Fin (n + 1) → { 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)) + (↑↑(σ ↑n) * (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
 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 + 1) → { 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)) +
         (↑↑(σ ↑n) * (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_131.{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 + 1) → { 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)) + (↑↑(σ ↑n) * (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
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 + 1) → { 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)) + (↑↑(σ ↑n) * (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
 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 + 1) → { 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.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 + 1) → { x // x ∈ {-1, 1} }
 a✝ : σ ∈ Finset.univ
 I : ι
 ⊢ Measurable fun a => ↑↑(σ ↑n) * (f I (X a.1.1) - f I (X a.1.2)) + c I]