zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_353.{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 :
    AEMeasurable (fun x => ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (x.2 k))|)
      ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))) :
  Measurable fun x => ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (x.2 k))| := sorry

Tactic state before:
case left.hf.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
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
⊢ AEMeasurable (fun x => ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (x.2 k))|)
    ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))

Tactic:
apply Measurable.aemeasurable

Tactic state after:
[case left.hf.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
 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
 ⊢ Measurable fun x => ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (x.2 k))|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_354.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) (k : Fin n)
  (h_original_goal : Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))) : Measurable fun a => ↑↑(i 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
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k : Fin n
⊢ Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))

Tactic:
apply Measurable.mul

Tactic state after:
[case 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
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => ↑↑(i k),
 case hg
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => f i_1 (X (a.2 k))]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_355.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) (k : Fin n)
  (h_original_goal : Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))) : Measurable fun a => f i_1 (X (a.2 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
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k : Fin n
⊢ Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))

Tactic:
apply Measurable.mul

Tactic state after:
[case 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
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => ↑↑(i k),
 case hg
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => f i_1 (X (a.2 k))]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_356.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) (k : Fin n) (h_original_goal : Measurable fun a => f i_1 (X (a.2 k))) :
  Measurable fun a => a.2 k := sorry

Tactic state before:
case hg
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k : Fin n
⊢ Measurable fun a => f i_1 (X (a.2 k))

Tactic:
apply (hf i_1).comp

Tactic state after:
[case hg
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => a.2 k]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_357.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) : Measurable Prod.snd := sorry

Tactic state before:
case hg.hg
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k : Fin n
⊢ Measurable Prod.snd

Tactic:
exact measurable_snd

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_358.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι)
  (h_original_goal : Measurable fun b => -∑ k, ↑↑(i k) * f i_1 (X (b.2 k))) :
  Measurable fun x => ∑ k, ↑↑(i k) * f i_1 (X (x.2 k)) := sorry

Tactic state before:
case left.hf.hf.h.hf.hf.hg
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
⊢ Measurable fun b => -∑ k, ↑↑(i k) * f i_1 (X (b.2 k))

Tactic:
apply Measurable.neg

Tactic state after:
[case left.hf.hf.h.hf.hf.hg.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
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 ⊢ Measurable fun x => ∑ k, ↑↑(i k) * f i_1 (X (x.2 k))]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_359.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) (k : Fin n)
  (h_original_goal : Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))) : Measurable fun a => ↑↑(i 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
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k : Fin n
⊢ Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))

Tactic:
apply Measurable.mul

Tactic state after:
[case 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
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => ↑↑(i k),
 case hg
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => f i_1 (X (a.2 k))]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_360.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) (k : Fin n)
  (h_original_goal : Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))) : Measurable fun a => f i_1 (X (a.2 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
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k : Fin n
⊢ Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))

Tactic:
apply Measurable.mul

Tactic state after:
[case 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
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => ↑↑(i k),
 case hg
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => f i_1 (X (a.2 k))]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_361.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) (k : Fin n) (h_original_goal : Measurable fun a => f i_1 (X (a.2 k))) :
  Measurable fun a => a.2 k := sorry

Tactic state before:
case hg
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k : Fin n
⊢ Measurable fun a => f i_1 (X (a.2 k))

Tactic:
apply (hf i_1).comp

Tactic state after:
[case hg
 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 : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => a.2 k]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_362.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) (k0 : ∀ (k : Fin n), Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))) :
  Measurable fun x => ∑ k, ↑↑(i k) * f i_1 (X (x.2 k)) := sorry

Tactic state before:
case left.hf.hf.h.hf.hf.hg.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
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 : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k0 : ∀ (k : Fin n), Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))
⊢ Measurable fun x => ∑ k, ↑↑(i k) * f i_1 (X (x.2 k))

Tactic:
exact Finset.measurable_sum Finset.univ fun i a ↦ k0 i

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_363.{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 :
    HasFiniteIntegral (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
      ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))) :
  ∀ᵐ (a : (Fin n → Ω) × (Fin n → Ω)) ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ),
    ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|‖ ≤ ↑n * b := sorry

Tactic state before:
case right
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
⊢ HasFiniteIntegral (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
    ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))

Tactic:
apply @MeasureTheory.hasFiniteIntegral_of_bounded _ _ _ _ _ _ _ (n * b)

Tactic state after:
[case right
 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
 ⊢ ∀ᵐ (a : (Fin n → Ω) × (Fin n → Ω)) ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ),
     ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|‖ ≤ ↑n * b]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_364.{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)
  (a : (Fin n → Ω) × (Fin n → Ω))
  (h_original_goal : ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|‖ ≤ ↑n * b) :
  (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ↑n * b := sorry

Tactic state before:
case right.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
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
a : (Fin n → Ω) × (Fin n → Ω)
⊢ ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|‖ ≤ ↑n * 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 right.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
 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
 a : (Fin n → Ω) × (Fin n → Ω)
 ⊢ (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ↑n * b]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_365.{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)
  (a : (Fin n → Ω) × (Fin n → Ω))
  (h_original_goal : (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ↑n * b) :
  |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ 2 ^ n * (↑n * b) := sorry

Tactic state before:
case right.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
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
a : (Fin n → Ω) × (Fin n → Ω)
⊢ (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ↑n * b

Tactic:
rw [inv_mul_le_iff₀]

Tactic state after:
[case right.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
 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
 a : (Fin n → Ω) × (Fin n → Ω)
 ⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ 2 ^ n * (↑n * b),
 case right.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
 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
 a : (Fin n → Ω) × (Fin n → Ω)
 ⊢ 0 < 2 ^ n]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_366.{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)
  (a : (Fin n → Ω) × (Fin n → Ω))
  (h_original_goal : (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ↑n * b) : 0 < 2 ^ n := sorry

Tactic state before:
case right.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
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
a : (Fin n → Ω) × (Fin n → Ω)
⊢ (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ↑n * b

Tactic:
rw [inv_mul_le_iff₀]

Tactic state after:
[case right.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
 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
 a : (Fin n → Ω) × (Fin n → Ω)
 ⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ 2 ^ n * (↑n * b),
 case right.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
 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
 a : (Fin n → Ω) × (Fin n → Ω)
 ⊢ 0 < 2 ^ n]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_367.{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)
  (a : (Fin n → Ω) × (Fin n → Ω)) :
  |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ∑ σ, |⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 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
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
a : (Fin n → Ω) × (Fin n → Ω)
⊢ |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ∑ σ, |⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))||

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_368.{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)
  (a : (Fin n → Ω) × (Fin n → Ω)) (h_original_goal : ∑ σ, |⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ∑ σ, ↑n * b)
  (i : Signs n) : i ∈ Finset.univ → |⨆ i_1, |∑ k, ↑↑(i k) * f i_1 (X (a.2 k))|| ≤ ↑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
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
a : (Fin n → Ω) × (Fin n → Ω)
⊢ ∑ σ, |⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|| ≤ ∑ σ, ↑n * b

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
 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
 a : (Fin n → Ω) × (Fin n → Ω)
 ⊢ ∀ i ∈ Finset.univ, |⨆ i_1, |∑ k, ↑↑(i k) * f i_1 (X (a.2 k))|| ≤ ↑n * b]