zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_210.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b)
  (this :
    (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
      ∫ (x : Fin n → Ω × Ω),
        (2 ^ n)⁻¹ * ∑ x_1, ⨆ I, ∑ i, ↑↑(x_1 i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ) :
  (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (2 ^ n)⁻¹ * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n : ℕ
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
this :
  (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (2 ^ n)⁻¹ * ∑ x_1, ⨆ I, ∑ i, ↑↑(x_1 i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ
⊢ (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (2 ^ n)⁻¹ * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ

Tactic:
exact this

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_211.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (a : ℝ)
  (ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a) ⦃x : ℝ⦄ (s : Fin 2) (i : ι)
  (eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x) (hax : |V (f i)| ≤ a)
  (h_original_goal : x ≤ a) : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) ≤ a := sorry

Tactic state before:
case h.intro.mk
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
a : ℝ
ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a
x : ℝ
s : Fin 2
i : ι
eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x
hax : |V (f i)| ≤ a
⊢ x ≤ a

Tactic:
rw [← eq]

Tactic state after:
[case h.intro.mk
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 a : ℝ
 ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a
 x : ℝ
 s : Fin 2
 i : ι
 eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x
 hax : |V (f i)| ≤ a
 ⊢ (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) ≤ a]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_212.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (a : ℝ)
  (ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a) ⦃x : ℝ⦄ (s : Fin 2) (i : ι)
  (eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x) (hax : |V (f i)| ≤ a)
  (h : (↑s == 0) = true) (h_original_goal : V (if (↑s == 0) = true then f i else -f i) ≤ a) : V (f i) ≤ a := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
a : ℝ
ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a
x : ℝ
s : Fin 2
i : ι
eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x
hax : |V (f i)| ≤ a
h : (↑s == 0) = true
⊢ V (if (↑s == 0) = true then f i else -f i) ≤ a

Tactic:
rw [if_pos h]

Tactic state after:
[Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 a : ℝ
 ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a
 x : ℝ
 s : Fin 2
 i : ι
 eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x
 hax : |V (f i)| ≤ a
 h : (↑s == 0) = true
 ⊢ V (f i) ≤ a]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_213.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (a : ℝ)
  (ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a) ⦃x : ℝ⦄ (s : Fin 2) (i : ι)
  (eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x) (hax : |V (f i)| ≤ a)
  (h : (↑s == 0) = true) : V (f i) ≤ a := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
a : ℝ
ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a
x : ℝ
s : Fin 2
i : ι
eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x
hax : |V (f i)| ≤ a
h : (↑s == 0) = true
⊢ V (f i) ≤ a

Tactic:
exact le_of_max_le_left hax

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_214.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (a : ℝ)
  (ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a) ⦃x : ℝ⦄ (s : Fin 2) (i : ι)
  (eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x) (hax : |V (f i)| ≤ a)
  (h : ¬(↑s == 0) = true) (h_original_goal : V (if (↑s == 0) = true then f i else -f i) ≤ a) : -V (f i) ≤ a := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
a : ℝ
ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a
x : ℝ
s : Fin 2
i : ι
eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x
hax : |V (f i)| ≤ a
h : ¬(↑s == 0) = true
⊢ V (if (↑s == 0) = true then f i else -f i) ≤ a

Tactic:
rw [if_neg h, hV₀]

Tactic state after:
[Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 a : ℝ
 ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a
 x : ℝ
 s : Fin 2
 i : ι
 eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x
 hax : |V (f i)| ≤ a
 h : ¬(↑s == 0) = true
 ⊢ -V (f i) ≤ a]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_215.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (a : ℝ)
  (ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a) ⦃x : ℝ⦄ (s : Fin 2) (i : ι)
  (eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x) (hax : |V (f i)| ≤ a)
  (h : ¬(↑s == 0) = true) : -V (f i) ≤ a := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
a : ℝ
ha : ∀ ⦃a_1 : ℝ⦄, (a_1 ∈ Set.range fun i => |V (f i)|) → a_1 ≤ a
x : ℝ
s : Fin 2
i : ι
eq : (fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (s, i) = x
hax : |V (f i)| ≤ a
h : ¬(↑s == 0) = true
⊢ -V (f i) ≤ a

Tactic:
exact le_of_max_le_right hax

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_216.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)))
  (h_original_goal : ⨆ i, |V (f i)| = ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) :
  ⨆ i, |V (f i)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
⊢ ⨆ i, |V (f i)| = ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)

Tactic:
apply le_antisymm

Tactic state after:
[case a
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 ⊢ ⨆ i, |V (f i)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2),
 case a
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 ⊢ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) ≤ ⨆ i, |V (f i)|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_217.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)))
  (h_original_goal : ⨆ i, |V (f i)| = ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) :
  ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) ≤ ⨆ i, |V (f i)| := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
⊢ ⨆ i, |V (f i)| = ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)

Tactic:
apply le_antisymm

Tactic state after:
[case a
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 ⊢ ⨆ i, |V (f i)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2),
 case a
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 ⊢ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) ≤ ⨆ i, |V (f i)|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_218.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)))
  (h_original_goal : ⨆ i, |V (f i)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) (x : ι) :
  |V (f x)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) := sorry

Tactic state before:
case a
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
⊢ ⨆ i, |V (f i)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)

Tactic:
apply ciSup_le

Tactic state after:
[case a.H
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 ⊢ ∀ (x : ι), |V (f x)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_219.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (i : ι)
  (h_original_goal : |V (f i)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) :
  V (f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) ∧
    -V (f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) := sorry

Tactic state before:
case a.H
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
i : ι
⊢ |V (f i)| ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)

Tactic:
apply abs_le'.mpr

Tactic state after:
[case a.H
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 i : ι
 ⊢ V (f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) ∧
     -V (f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_220.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (i : ι) :
  V (f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) := sorry

Tactic state before:
case a.H.left
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
i : ι
⊢ V (f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)

Tactic:
exact le_ciSup hV₁' ⟨(0 : Fin 2), i⟩

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_221.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (i : ι)
  (h_original_goal : -V (f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)) :
  V (-f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) := sorry

Tactic state before:
case a.H.right
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
i : ι
⊢ -V (f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)

Tactic:
rw [← hV₀]

Tactic state after:
[case a.H.right
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 i : ι
 ⊢ V (-f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_222.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (i : ι) :
  V (-f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) := sorry

Tactic state before:
case a.H.right
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
i : ι
⊢ V (-f i) ≤ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)

Tactic:
exact le_ciSup hV₁' ⟨(1 : Fin 2), i⟩

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_223.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2)))
  (h_original_goal : ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) ≤ ⨆ i, |V (f i)|) (x : Fin 2 × ι) :
  V (if (↑x.1 == 0) = true then f x.2 else -f x.2) ≤ ⨆ i, |V (f i)| := sorry

Tactic state before:
case a
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
⊢ ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) ≤ ⨆ i, |V (f i)|

Tactic:
apply ciSup_le

Tactic state after:
[case a.H
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 ⊢ ∀ (x : Fin 2 × ι), V (if (↑x.1 == 0) = true then f x.2 else -f x.2) ≤ ⨆ i, |V (f i)|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_224.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (s : Fin 2) (i : ι)
  (h_original_goal : V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ ⨆ i, |V (f i)|) :
  V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)| := sorry

Tactic state before:
case a.H.mk
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
s : Fin 2
i : ι
⊢ V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ ⨆ i, |V (f i)|

Tactic:
apply le_trans _ (le_ciSup hV₁ i)

Tactic state after:
[Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 s : Fin 2
 i : ι
 ⊢ V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_225.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (s : Fin 2) (i : ι)
  (h : (↑s == 0) = true)
  (h_original_goal : V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)|) :
  V (f (s, i).2) ≤ |V (f i)| := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
s : Fin 2
i : ι
h : (↑s == 0) = true
⊢ V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)|

Tactic:
rw [if_pos h]

Tactic state after:
[Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 s : Fin 2
 i : ι
 h : (↑s == 0) = true
 ⊢ V (f (s, i).2) ≤ |V (f i)|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_226.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (s : Fin 2) (i : ι)
  (h : (↑s == 0) = true) : V (f (s, i).2) ≤ |V (f i)| := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
s : Fin 2
i : ι
h : (↑s == 0) = true
⊢ V (f (s, i).2) ≤ |V (f i)|

Tactic:
exact le_abs_self (V (f i))

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_227.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (s : Fin 2) (i : ι)
  (h : ¬(↑s == 0) = true)
  (h_original_goal : V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)|) :
  -V (f (s, i).2) ≤ |V (f i)| := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
s : Fin 2
i : ι
h : ¬(↑s == 0) = true
⊢ V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)|

Tactic:
rw [if_neg h, hV₀]

Tactic state after:
[Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 s : Fin 2
 i : ι
 h : ¬(↑s == 0) = true
 ⊢ -V (f (s, i).2) ≤ |V (f i)|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_228.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (s : Fin 2) (i : ι)
  (h : ¬(↑s == 0) = true) : -V (f (s, i).2) ≤ |V (f i)| := sorry

Tactic state before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
s : Fin 2
i : ι
h : ¬(↑s == 0) = true
⊢ -V (f (s, i).2) ≤ |V (f i)|

Tactic:
exact neg_le_abs (V (f i))

Tactic state after: