zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_275.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ) (i : ι)
  (h_original_goal : |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|) :
  BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|) := sorry

Tactic state before:
case h.hf
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
i : ι
⊢ |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|

Tactic:
apply le_ciSup_of_le

Tactic state after:
[case h.hf.H
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 i : ι
 ⊢ BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|),
 case h.hf.h
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 i : ι
 ⊢ |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ |∑ k, ↑↑(σ k) * f ?h.hf.c (X (ω.1 k))|,
 case h.hf.c
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 i : ι
 ⊢ ι]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_276.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ) :
  BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|) := sorry

Tactic state before:
case h.hf.H
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
i : ι
⊢ BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|)

Tactic:
apply bdd_above_aux' hf'

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_277.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ) (i : ι) :
  |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| := sorry

Tactic state before:
case h.hf.h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
i : ι
⊢ |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ |∑ k, ↑↑(σ k) * f ?h.hf.c (X (ω.1 k))|

Tactic:
apply le_of_eq rfl

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_278.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω))
  (h_original_goal : ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ∑ σ, ⨆ i, ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|)
  (i : Signs n) :
  i ∈ Finset.univ → ⨆ i_1, |∑ k, ↑↑(i k) * f i_1 (X (ω.1 k))| ≤ ⨆ i_1, ∑ k, |↑↑(i k) * f i_1 (X (ω.1 k))| := sorry

Tactic state before:
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
⊢ ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ∑ σ, ⨆ i, ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|

Tactic:
apply Finset.sum_le_sum

Tactic state after:
[case h
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 ⊢ ∀ i ∈ Finset.univ, ⨆ i_1, |∑ k, ↑↑(i k) * f i_1 (X (ω.1 k))| ≤ ⨆ i_1, ∑ k, |↑↑(i k) * f i_1 (X (ω.1 k))|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_279.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ)
  (h_original_goal : ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ⨆ i, ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|) :
  BddAbove (Set.range fun i => ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|) := sorry

Tactic state before:
case h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ⨆ i, ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|

Tactic:
apply ciSup_mono

Tactic state after:
[case h.B
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ BddAbove (Set.range fun i => ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|),
 case h.H
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ ∀ (x : ι), |∑ k, ↑↑(σ k) * f x (X (ω.1 k))| ≤ ∑ k, |↑↑(σ k) * f x (X (ω.1 k))|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_280.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ)
  (h_original_goal : ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ⨆ i, ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|) (x : ι) :
  |∑ k, ↑↑(σ k) * f x (X (ω.1 k))| ≤ ∑ k, |↑↑(σ k) * f x (X (ω.1 k))| := sorry

Tactic state before:
case h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ⨆ i, ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|

Tactic:
apply ciSup_mono

Tactic state after:
[case h.B
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ BddAbove (Set.range fun i => ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|),
 case h.H
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ ∀ (x : ι), |∑ k, ↑↑(σ k) * f x (X (ω.1 k))| ≤ ∑ k, |↑↑(σ k) * f x (X (ω.1 k))|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_281.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ) :
  BddAbove (Set.range fun i => ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|) := sorry

Tactic state before:
case h.B
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ BddAbove (Set.range fun i => ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|)

Tactic:
apply bdd_above_aux hf'

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_282.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ) (i : ι) :
  |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ∑ k, |↑↑(σ k) * f i (X (ω.1 k))| := sorry

Tactic state before:
case h.H
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
i : ι
⊢ |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|

Tactic:
exact Finset.abs_sum_le_sum_abs (fun i_1 ↦ ↑↑(σ i_1) * f i (X (ω.1 i_1))) Finset.univ

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_283.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (i : ι) (k : Fin n) :
  |↑↑(σ k) * f i (X (ω.1 k))| = |f i (X (ω.1 k))| := sorry

Tactic state before:
case e_f.h.e_s.h.e_f.h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
i : ι
k : Fin n
⊢ |↑↑(σ k) * f i (X (ω.1 k))| = |f i (X (ω.1 k))|

Tactic:
simp only [Int.reduceNeg, abs_mul, Signs.apply_abs', one_mul]

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_284.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ)
  (h_original_goal : ⨆ i, ∑ k, |f i (X (ω.1 k))| ≤ ⨆ i, ∑ k, b) : BddAbove (Set.range fun i => ∑ k, b) := sorry

Tactic state before:
case h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ ⨆ i, ∑ k, |f i (X (ω.1 k))| ≤ ⨆ i, ∑ k, b

Tactic:
apply ciSup_mono

Tactic state after:
[case h.B
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ BddAbove (Set.range fun i => ∑ k, b),
 case h.H
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ ∀ (x : ι), ∑ k, |f x (X (ω.1 k))| ≤ ∑ k, b]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_285.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ)
  (h_original_goal : ⨆ i, ∑ k, |f i (X (ω.1 k))| ≤ ⨆ i, ∑ k, b) (x : ι) : ∑ k, |f x (X (ω.1 k))| ≤ ∑ k, b := sorry

Tactic state before:
case h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ ⨆ i, ∑ k, |f i (X (ω.1 k))| ≤ ⨆ i, ∑ k, b

Tactic:
apply ciSup_mono

Tactic state after:
[case h.B
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ BddAbove (Set.range fun i => ∑ k, b),
 case h.H
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ ∀ (x : ι), ∑ k, |f x (X (ω.1 k))| ≤ ∑ k, b]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_286.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (σ : Signs n) (a : σ ∈ Finset.univ) :
  BddAbove (Set.range fun i => ∑ k, b) := sorry

Tactic state before:
case h.B
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ BddAbove (Set.range fun i => ∑ k, b)

Tactic:
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, Set.range_const,
  bddAbove_singleton]

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_287.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ) (i : ι)
  (k : Fin n) (a_1 : k ∈ Finset.univ) : |f i (X (ω.1 k))| ≤ b := sorry

Tactic state before:
case h.H.h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝¹ : σ ∈ Finset.univ
i : ι
k : Fin n
a✝ : k ∈ Finset.univ
⊢ |f i (X (ω.1 k))| ≤ b

Tactic:
exact hf' i (X (ω.1 k))

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_288.{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 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) : ∑ σ, ⨆ i, ∑ k, b ≤ 2 ^ n * (↑n * b) := sorry

Tactic state before:
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
⊢ ∑ σ, ⨆ i, ∑ k, b ≤ 2 ^ n * (↑n * b)

Tactic:
simp

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_289.{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] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (h_original_goal :
    Integrable (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ)
      (Measure.pi fun x => μ)) :
  MemLp (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ) 1
    (Measure.pi fun x => μ) := sorry

Tactic state before:
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
X : Ω → Z
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
⊢ Integrable (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ)
    (Measure.pi fun x => μ)

Tactic:
simp_rw [← memLp_one_iff_integrable]

Tactic state after:
[Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ⊢ MemLp (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ) 1
     (Measure.pi fun x => μ)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_290.{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] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (h_original_goal :
    AEStronglyMeasurable
      (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ)
      (Measure.pi fun x => μ))
  (i : ι) :
  Measurable fun b => ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (b k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ := sorry

Tactic state before:
case refine_1
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
X : Ω → Z
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
⊢ AEStronglyMeasurable
    (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ)
    (Measure.pi fun x => μ)

Tactic:
apply (Measurable.iSup _).aestronglyMeasurable

Tactic state after:
[Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ⊢ ∀ (i : ι), Measurable fun b => ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (b k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ]