zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_142.{u_1, w, u} {Ω : Type u} {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  [inst : Countable ι] {X : Ω → Z} {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω) (σ : Signs n)
  (this : ∃ x, ∀ y ∈ Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ x) (z : ℝ)
  (hz : ∀ y ∈ Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ z) (y : ℝ)
  (hy : ∃ y_1, |∑ k, ↑↑(σ k) * (f y_1 (X (ω k).1) - f y_1 (X (ω k).2))| = y) (i : ι)
  (hi : |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| = y) :
  ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| ∈
    Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| := sorry

Tactic state before:
case a
Ω : Type u
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
inst✝ : Countable ι
X : Ω → Z
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
σ : Signs n
this : ∃ x, ∀ y ∈ Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ x
z : ℝ
hz : ∀ y ∈ Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ z
y : ℝ
hy : ∃ y_1, |∑ k, ↑↑(σ k) * (f y_1 (X (ω k).1) - f y_1 (X (ω k).2))| = y
i : ι
hi : |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| = y
⊢ ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| ∈
    Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|

Tactic:
simp

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_143.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (h_original_goal :
    Integrable (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)
      (Measure.pi fun x => μ.prod μ)) :
  MemLp (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) 1
    (Measure.pi fun x => μ.prod μ) := sorry

Tactic state before:
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
⊢ Integrable (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)
    (Measure.pi fun x => μ.prod μ)

Tactic:
rw [← memLp_one_iff_integrable]

Tactic state after:
[Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ⊢ MemLp (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) 1
     (Measure.pi fun x => μ.prod μ)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_144.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (h_original_goal :
    MemLp (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) 1
      (Measure.pi fun x => μ.prod μ)) :
  ∀ᵐ (x : Fin n → Ω × Ω) ∂Measure.pi fun x => μ.prod μ,
    ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (x k).1) - f i (X (x k).2))|‖ ≤
      ↑n * (b + b) := sorry

Tactic state before:
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
⊢ MemLp (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) 1
    (Measure.pi fun x => μ.prod μ)

Tactic:
apply MemLp.of_bound (measurable_aux3 hf).aestronglyMeasurable (n * (b + b))

Tactic state after:
[Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ⊢ ∀ᵐ (x : Fin n → Ω × Ω) ∂Measure.pi fun x => μ.prod μ,
     ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (x k).1) - f i (X (x k).2))|‖ ≤ ↑n * (b + b)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_145.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (h_original_goal :
    ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|‖ ≤ ↑n * (b + b)) :
  (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ ↑n * (b + b) := sorry

Tactic state before:
case h
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
⊢ ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|‖ ≤ ↑n * (b + b)

Tactic:
simp only [Signs.card, Nat.cast_pow, Nat.cast_ofNat, Int.reduceNeg, norm_mul, norm_inv, norm_pow,
  norm_ofNat, norm_eq_abs]

Tactic state after:
[case h
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ⊢ (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ ↑n * (b + b)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_146.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (h_original_goal : (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ ↑n * (b + b)) :
  |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ ↑n * (b + b) * 2 ^ n := sorry

Tactic state before:
case h
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
⊢ (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ ↑n * (b + b)

Tactic:
rw [inv_mul_le_iff₀' (by simp)]

Tactic state after:
[case h
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 ⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ ↑n * (b + b) * 2 ^ n]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_147.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (h_original_goal : |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ ↑n * (b + b) * 2 ^ n) :
  |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ 2 ^ n * (↑n * (b + b)) := sorry

Tactic state before:
case h
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ ↑n * (b + b) * 2 ^ n

Tactic:
rw [mul_comm (↑n * (b + b))]

Tactic state after:
[case h
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 ⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤ 2 ^ n * (↑n * (b + b))]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_148.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω : Fin n → Ω × Ω) :
  |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
    ∑ σ, |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| := sorry

Tactic state before:
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
    ∑ σ, |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))||

Tactic:
apply Finset.abs_sum_le_sum_abs

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_149.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (h_original_goal :
    ∑ σ, |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
      ∑ σ, ⨆ i, |(|∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)|)
  (i : Signs n) :
  i ∈ Finset.univ →
    |⨆ i_1, |∑ k, ↑↑(i k) * (f i_1 (X (ω k).1) - f i_1 (X (ω k).2))|| ≤
      ⨆ i_1, |(|∑ k, ↑↑(i k) * (f i_1 (X (ω k).1) - f i_1 (X (ω k).2))|)| := sorry

Tactic state before:
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
⊢ ∑ σ, |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
    ∑ σ, ⨆ i, |(|∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)|

Tactic:
apply Finset.sum_le_sum

Tactic state after:
[case h
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 ⊢ ∀ i ∈ Finset.univ,
     |⨆ i_1, |∑ k, ↑↑(i k) * (f i_1 (X (ω k).1) - f i_1 (X (ω k).2))|| ≤
       ⨆ i_1, |(|∑ k, ↑↑(i k) * (f i_1 (X (ω k).1) - f i_1 (X (ω k).2))|)|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_150.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (σ : Signs n) (a : σ ∈ Finset.univ)
  (h_original_goal :
    |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
      ⨆ i, |(|∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)|) :
  |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
    ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| := sorry

Tactic state before:
case h
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
    ⨆ i, |(|∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)|

Tactic:
simp only [Int.reduceNeg, abs_abs]

Tactic state after:
[case h
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
     ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_151.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (σ : Signs n) (a : σ ∈ Finset.univ)
  (h_original_goal :
    |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
      ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) :
  |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| =
    ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| := sorry

Tactic state before:
case h
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| ≤
    ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|

Tactic:
apply Eq.le

Tactic state after:
[case h.hab
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| =
     ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_152.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (σ : Signs n) (a : σ ∈ Finset.univ)
  (h_original_goal :
    |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| =
      ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) :
  0 ≤ ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| := sorry

Tactic state before:
case h.hab
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ |⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|| =
    ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|

Tactic:
simp only [Int.reduceNeg, abs_eq_self]

Tactic state after:
[case h.hab
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ 0 ≤ ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_153.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (σ : Signs n) (a : σ ∈ Finset.univ) (h : Nonempty ι)
  (h_original_goal : 0 ≤ ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) :
  0 ≤ |∑ k, ↑↑(σ k) * (f (Classical.choice h) (X (ω k).1) - f (Classical.choice h) (X (ω k).2))| := sorry

Tactic state before:
case pos
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
σ : Signs n
a✝ : σ ∈ Finset.univ
h : Nonempty ι
⊢ 0 ≤ ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|

Tactic:
apply le_ciSup_of_le _ (Classical.choice h)

Tactic state after:
[case pos
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 h : Nonempty ι
 ⊢ 0 ≤ |∑ k, ↑↑(σ k) * (f (Classical.choice h) (X (ω k).1) - f (Classical.choice h) (X (ω k).2))|,
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 h : Nonempty ι
 ⊢ BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_154.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (σ : Signs n) (a : σ ∈ Finset.univ) (h : Nonempty ι)
  (h_original_goal : 0 ≤ ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) :
  BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|) := sorry

Tactic state before:
case pos
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
σ : Signs n
a✝ : σ ∈ Finset.univ
h : Nonempty ι
⊢ 0 ≤ ⨆ i, |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|

Tactic:
apply le_ciSup_of_le _ (Classical.choice h)

Tactic state after:
[case pos
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 h : Nonempty ι
 ⊢ 0 ≤ |∑ k, ↑↑(σ k) * (f (Classical.choice h) (X (ω k).1) - f (Classical.choice h) (X (ω k).2))|,
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω × Ω
 ω₁ : Fin n → Ω := fun k => (ω k).1
 ω₂ : Fin n → Ω := fun k => (ω k).2
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 h : Nonempty ι
 ⊢ BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_155.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω)
  (σ : Signs n) (a : σ ∈ Finset.univ) (h : Nonempty ι) :
  0 ≤ |∑ k, ↑↑(σ k) * (f (Classical.choice h) (X (ω k).1) - f (Classical.choice h) (X (ω k).2))| := sorry

Tactic state before:
case pos
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
ω₁ : Fin n → Ω := fun k => (ω k).1
ω₂ : Fin n → Ω := fun k => (ω k).2
σ : Signs n
a✝ : σ ∈ Finset.univ
h : Nonempty ι
⊢ 0 ≤ |∑ k, ↑↑(σ k) * (f (Classical.choice h) (X (ω k).1) - f (Classical.choice h) (X (ω k).2))|

Tactic:
simp

Tactic state after: