awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
⊢ LE.le (α := ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have refine : ∀ (I : Finset ℕ),
  LE.le (α := ℝ≥0∞) (∑ i ∈ I, (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
Tactic: refine ENNReal.summable.tsum_le_of_sum_le fun I ↦ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ I, (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have rw : LE.le (α := ℝ≥0∞) (∑ s ∈ Finset.image f I, (m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
  ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
Tactic: rw [← Finset.sum_image_of_disjoint addContent_empty (hf_disj.pairwiseDisjoint _)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ LE.le (α := ℝ≥0∞) (∑ s ∈ Finset.image f I, (m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have refine_1 : (↑(Finset.image f I) : Set (Set α)) ⊆ C
Tactic: refine sum_addContent_le_of_subset hC (I := I.image f) ?_ ?_ hf_Union ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
refine_1 : (↑(Finset.image f I) : Set (Set α)) ⊆ C
⊢ LE.le (α := ℝ≥0∞) (∑ s ∈ Finset.image f I, (m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have refine_2 : (↑(Finset.image f I) : Set (Set α)).PairwiseDisjoint id
Tactic: refine sum_addContent_le_of_subset hC (I := I.image f) ?_ ?_ hf_Union ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
refine_1 : (↑(Finset.image f I) : Set (Set α)) ⊆ C
refine_2 : (↑(Finset.image f I) : Set (Set α)).PairwiseDisjoint id
⊢ LE.le (α := ℝ≥0∞) (∑ s ∈ Finset.image f I, (m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have refine_3 : ∀ s ∈ Finset.image f I, s ⊆ ⋃ (i : ℕ), f i
Tactic: refine sum_addContent_le_of_subset hC (I := I.image f) ?_ ?_ hf_Union ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ (↑(Finset.image f I) : Set (Set α)) ⊆ C
have refine_1 : (↑I : Set ℕ) ⊆ f ⁻¹' C
Tactic: simp only [coe_image, Set.image_subset_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ (↑I : Set ℕ) ⊆ f ⁻¹' C
have refine_1 : f '' (↑I : Set ℕ) ⊆ C
Tactic: refine (subset_preimage_image f I).trans (preimage_mono ?_)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ (↑(Finset.image f I) : Set (Set α)).PairwiseDisjoint id
have refine_2 : (f '' (↑I : Set ℕ)).PairwiseDisjoint id
Tactic: simp only [coe_image]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
s : Set α
hs : s ∈ f '' (↑I : Set ℕ)
t : Set α
ht : t ∈ f '' (↑I : Set ℕ)
hst : s ≠ t
⊢ (Disjoint on id) s t
have refine_2 : ∀ (hs : ∃ x ∈ (↑I : Set ℕ), f x = s) ⦃t : Set α⦄ (ht : ∃ x ∈ (↑I : Set ℕ), f x = t) (hst : s ≠ t), (Disjoint on id) s t
Tactic: rw [Set.mem_image] at hs ht
Cleared hypotheses: hs, t, ht, hst
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
s : Set α
hs : ∃ x ∈ (↑I : Set ℕ), f x = s
t : Set α
ht : ∃ x ∈ (↑I : Set ℕ), f x = t
hst : s ≠ t
⊢ (Disjoint on id) s t
have refine_2 : ∀ i ∈ (↑I : Set ℕ), ∀ (hst : f i ≠ t), (Disjoint on id) (f i) t
Tactic: obtain ⟨i, _, rfl⟩ := hs
Cleared hypotheses: s, hs, hst
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
t : Set α
ht : ∃ x ∈ (↑I : Set ℕ), f x = t
i : ℕ
left : i ∈ (↑I : Set ℕ)
hst : f i ≠ t
⊢ (Disjoint on id) (f i) t
have refine_2 : ∀ j ∈ (↑I : Set ℕ), ∀ (hst : f i ≠ f j), (Disjoint on id) (f i) (f j)
Tactic: obtain ⟨j, _, rfl⟩ := ht
Cleared hypotheses: t, ht, hst
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (hypothesis)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
i : ℕ
left : i ∈ (↑I : Set ℕ)
j : ℕ
left_1 : j ∈ (↑I : Set ℕ)
hst : f i ≠ f j
h_eq : i = j
⊢ False
have hst₁ : f j ≠ f j
Tactic: rw [h_eq] at hst
Cleared hypotheses: hst, h_eq
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (hypothesis)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
i : ℕ
left : i ∈ (↑I : Set ℕ)
j : ℕ
left_1 : j ∈ (↑I : Set ℕ)
hst : f i ≠ f j
⊢ (Disjoint on id) (f i) (f j)
have hij : i ≠ j
Tactic: have hij : i ≠ j := by intro h_eq; rw [h_eq] at hst; exact hst rfl
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ ∀ s ∈ Finset.image f I, s ⊆ ⋃ (i : ℕ), f i
have refine_3 : ∀ a ∈ I, f a ⊆ ⋃ (i : ℕ), f i
Tactic: simp only [Finset.mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ I, (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have rw : LE.le (α := ℝ≥0∞) (∑ s ∈ Finset.image f I, (m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
  ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
Tactic: classical
  rw [← Finset.sum_image_of_disjoint addContent_empty (hf_disj.pairwiseDisjoint _)]
  refine sum_addContent_le_of_subset hC (I := I.image f) ?_ ?_ hf_Union ?_
  · simp only [coe_image, Set.image_subset_iff]
    refine (subset_preimage_image f I).trans (preimage_mono ?_)
    rintro i ⟨j, _, rfl⟩
    exact hf j
  · simp only [coe_image]
    intro s hs t ht hst
    rw [Set.mem_image] at hs ht
    obtain ⟨i, _, rfl⟩ := hs
    obtain ⟨j, _, rfl⟩ := ht
    have hij : i ≠ j := by intro h_eq; rw [h_eq] at hst; exact hst rfl
    exact hf_disj hij
  · simp only [Finset.mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
    exact fun i _ ↦ subset_iUnion _ i
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
rw rw₁ rw₂ :
  LE.le (α := ℝ≥0∞) (∑ s ∈ Finset.image f I, (m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ I, (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have refine_1 : (↑(Finset.image f I) : Set (Set α)) ⊆ C
Tactic: classical
  rw [← Finset.sum_image_of_disjoint addContent_empty (hf_disj.pairwiseDisjoint _)]
  refine sum_addContent_le_of_subset hC (I := I.image f) ?_ ?_ hf_Union ?_
  · simp only [coe_image, Set.image_subset_iff]
    refine (subset_preimage_image f I).trans (preimage_mono ?_)
    rintro i ⟨j, _, rfl⟩
    exact hf j
  · simp only [coe_image]
    intro s hs t ht hst
    rw [Set.mem_image] at hs ht
    obtain ⟨i, _, rfl⟩ := hs
    obtain ⟨j, _, rfl⟩ := ht
    have hij : i ≠ j := by intro h_eq; rw [h_eq] at hst; exact hst rfl
    exact hf_disj hij
  · simp only [Finset.mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
    exact fun i _ ↦ subset_iUnion _ i
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
rw rw₁ rw₂ :
  LE.le (α := ℝ≥0∞) (∑ s ∈ Finset.image f I, (m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
refine_1 : (↑(Finset.image f I) : Set (Set α)) ⊆ C
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ I, (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have refine_2 : (↑(Finset.image f I) : Set (Set α)).PairwiseDisjoint id
Tactic: classical
  rw [← Finset.sum_image_of_disjoint addContent_empty (hf_disj.pairwiseDisjoint _)]
  refine sum_addContent_le_of_subset hC (I := I.image f) ?_ ?_ hf_Union ?_
  · simp only [coe_image, Set.image_subset_iff]
    refine (subset_preimage_image f I).trans (preimage_mono ?_)
    rintro i ⟨j, _, rfl⟩
    exact hf j
  · simp only [coe_image]
    intro s hs t ht hst
    rw [Set.mem_image] at hs ht
    obtain ⟨i, _, rfl⟩ := hs
    obtain ⟨j, _, rfl⟩ := ht
    have hij : i ≠ j := by intro h_eq; rw [h_eq] at hst; exact hst rfl
    exact hf_disj hij
  · simp only [Finset.mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
    exact fun i _ ↦ subset_iUnion _ i
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
rw rw₁ rw₂ :
  LE.le (α := ℝ≥0∞) (∑ s ∈ Finset.image f I, (m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
refine_1 : (↑(Finset.image f I) : Set (Set α)) ⊆ C
refine_2 : (↑(Finset.image f I) : Set (Set α)).PairwiseDisjoint id
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ I, (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
have refine_3 : ∀ s ∈ Finset.image f I, s ⊆ ⋃ (i : ℕ), f i
Tactic: classical
  rw [← Finset.sum_image_of_disjoint addContent_empty (hf_disj.pairwiseDisjoint _)]
  refine sum_addContent_le_of_subset hC (I := I.image f) ?_ ?_ hf_Union ?_
  · simp only [coe_image, Set.image_subset_iff]
    refine (subset_preimage_image f I).trans (preimage_mono ?_)
    rintro i ⟨j, _, rfl⟩
    exact hf j
  · simp only [coe_image]
    intro s hs t ht hst
    rw [Set.mem_image] at hs ht
    obtain ⟨i, _, rfl⟩ := hs
    obtain ⟨j, _, rfl⟩ := ht
    have hij : i ≠ j := by intro h_eq; rw [h_eq] at hst; exact hst rfl
    exact hf_disj hij
  · simp only [Finset.mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
    exact fun i _ ↦ subset_iUnion _ i
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (hypothesis)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
u v : α
h : u ≤ v
⊢ onIocAux f (Set.Ioc u v) = f v - f u
have h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2
Tactic: have h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Ioc u v = Ioc p.1 p.2 := ⟨(u, v), h, rfl⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
u v : α
h : u ≤ v
h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2
⊢ onIocAux f (Set.Ioc u v) = f v - f u
have simp : f sorry.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2 -
    f sorry.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).1 =
  f v - f u
Tactic: simp only [onIocAux, h', ↓reduceDIte]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (hypothesis)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
u v : α
h : u ≤ v
h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2
u' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).1
v' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2
⊢ f v' - f u' = f v - f u
have hu'v' : u' ≤ v' ∧ Set.Ioc u v = Set.Ioc u' v'
Tactic: have hu'v' : u' ≤ v' ∧ Ioc u v = Ioc u' v' := h'.choose_spec
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
u v : α
h : u ≤ v
h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2
u' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).1
v' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2
hu'v' : u' ≤ v' ∧ Set.Ioc u v = Set.Ioc u' v'
⊢ f v' - f u' = f v - f u
have inl : ∀ (h : u ≤ u) (h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2),
  let u' := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2).1;
  let v' := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2).2;
  ∀ (hu'v' : u' ≤ v' ∧ Set.Ioc u u = Set.Ioc u' v'), f v' - f u' = f u - f u
Tactic: rcases h.eq_or_lt with rfl | huv
Cleared hypotheses: v, h, h', u', v', hu'v'
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
u v : α
h : u ≤ v
h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2
u' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).1
v' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2
hu'v' : u' ≤ v' ∧ Set.Ioc u v = Set.Ioc u' v'
inl :
  ∀ (h : u ≤ u) (h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2),
    let u' := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2).1;
    let v' := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2).2;
    ∀ (hu'v' : u' ≤ v' ∧ Set.Ioc u u = Set.Ioc u' v'), f v' - f u' = f u - f u
⊢ f v' - f u' = f v - f u
have inr : ∀ (huv : u < v), f v' - f u' = f v - f u
Tactic: rcases h.eq_or_lt with rfl | huv
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (hypothesis)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
u v : α
h : u ≤ v
h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2
u' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).1
v' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2
hu'v' : u' ≤ v' ∧ Set.Ioc u v = Set.Ioc u' v'
huv : u < v
⊢ f v' - f u' = f v - f u
have hu'v'₁ : u' ≤ v' ∧ u = u' ∧ v = v'
Tactic: rw [Ioc_eq_Ioc_iff (Or.inl huv)] at hu'v'
Cleared hypotheses: hu'v', huv
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
⊢ onIocAux f ∅ = 0
have rw : ∀ (h : ∃ (p : α × α), p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2),
  f h.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2).2 -
      f h.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2).1 =
    0
Tactic: rw [onIocAux, dite_eq_right_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
⊢ onIocAux f ∅ = 0
have rw : Eq.{u_3 + 1} (α := G)
  (if h : ∃ (p : α × α), p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2 then
    f h.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2).2 -
      f h.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2).1
  else 0)
  0
Tactic: classical
  rw [onIocAux, dite_eq_right_iff]
  grind [Set.Ioc_eq_empty_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
rw :
  Eq.{u_3 + 1} (α := G)
    (if h : ∃ (p : α × α), p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2 then
      f h.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2).2 -
        f h.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2).1
    else 0)
    0
⊢ onIocAux f ∅ = 0
have rw₁ : ∀ (h : ∃ (p : α × α), p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2),
  f h.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2).2 -
      f h.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ ∅ = Set.Ioc p.1 p.2).1 =
    0
Tactic: classical
  rw [onIocAux, dite_eq_right_iff]
  grind [Set.Ioc_eq_empty_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (hypothesis)
Goal:
α : Type u_1
C : Set (Set α)
s t : Set α
I_1 : Finset (Set α)
G_1 : Type u_2
inst : AddCommMonoid G_1
m m' : AddContent G_1 C
inst_1 : LinearOrder α
G : Type u_3
inst_2 : AddCommGroup G
f : α → G
I : Finset (Set α)
hI : (↑I : Set (Set α)) ⊆ {s : Set α | ∃ (u : α) (v : α), u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I : Set (Set α)).PairwiseDisjoint id
h''I :
  Membership.mem (γ := Set (Set α)) {s : Set α | ∃ (u : α) (v : α), u ≤ v ∧ s = Set.Ioc u v} (⋃₀ (↑I : Set (Set α)))
hn : #I = 0
⊢ onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
have this : I = ∅
Tactic: have : I = ∅ := by grind