awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
    ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i)
have simp_rw : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
  ∑' (i : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
      (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
Tactic: simp_rw [h_m_eq]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
    ∑' (i : ℕ),
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
have rw : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
  HAdd.hAdd (α := ℝ≥0∞) (∑' (a : ℕ), (m : (a : Set α) → ℝ≥0∞) (f a ∩ s))
    (∑' (a : ℕ), ∑ u ∈ A a, (m : (a : Set α) → ℝ≥0∞) u)
Tactic: rw [ENNReal.tsum_add]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
    HAdd.hAdd (α := ℝ≥0∞) (∑' (a : ℕ), (m : (a : Set α) → ℝ≥0∞) (f a ∩ s))
      (∑' (a : ℕ), ∑ u ∈ A a, (m : (a : Set α) → ℝ≥0∞) u)
have refine_1 : LE.le (α := ℝ≥0∞)
  (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
  (∑' (a : ℕ), (m : (a : Set α) → ℝ≥0∞) (f a ∩ s) : ℝ≥0∞)
Tactic: refine add_le_add ?_ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
refine_1 :
  LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
    (∑' (a : ℕ), (m : (a : Set α) → ℝ≥0∞) (f a ∩ s) : ℝ≥0∞)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
    HAdd.hAdd (α := ℝ≥0∞) (∑' (a : ℕ), (m : (a : Set α) → ℝ≥0∞) (f a ∩ s))
      (∑' (a : ℕ), ∑ u ∈ A a, (m : (a : Set α) → ℝ≥0∞) u)
have refine_2 : LE.le (α := ℝ≥0∞)
  (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞)
  (∑' (a : ℕ), ∑ u ∈ A a, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
Tactic: refine add_le_add ?_ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
    (∑' (a : ℕ), (m : (a : Set α) → ℝ≥0∞) (f a ∩ s) : ℝ≥0∞)
have refine_1 : t ∩ s ⊆ ⋃ (i : ℕ), (fun (i : ℕ) ↦ f i ∩ s) i
Tactic: refine iInf_le_of_le (fun i ↦ f i ∩ s) <| iInf_le_of_le ?_ le_rfl
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ t ∩ s ⊆ ⋃ (i : ℕ), (fun (i : ℕ) ↦ f i ∩ s) i
have refine_1 : t ∩ s ⊆ (⋃ (i : ℕ), f i) ∩ s
Tactic: rw [← iUnion_inter]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞)
    (∑' (a : ℕ), ∑ u ∈ A a, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
have refine_2 : OuterMeasure.measureOf (α := α) (OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry) (⋃ (i : ℕ), f i \ s) ≤
  ∑' (a : ℕ), ∑ u ∈ A a, (m : (a : Set α) → ℝ≥0∞) u
Tactic: apply le_trans <| (OuterMeasure.ofFunction m addContent_empty).mono
      <| (iUnion_diff s f) ▸ diff_subset_diff_left hf_subset
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ OuterMeasure.measureOf (α := α) (OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯) (⋃ (i : ℕ), f i \ s) ≤
    ∑' (a : ℕ), ∑ u ∈ A a, (m : (a : Set α) → ℝ≥0∞) u
have refine_2 : LE.le (α := ℝ≥0∞)
  (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i \ s) :
    ℝ≥0∞)
  (∑' (a : ℕ), ∑ u ∈ hC.disjointOfDiff (C := C) (s := f a) (t := f a ∩ s) sorry sorry, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
Tactic: simp only [OuterMeasure.measureOf_eq_coe, A]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞)
        (⋃ (i : ℕ), f i \ s) :
      ℝ≥0∞)
    (∑' (a : ℕ), ∑ u ∈ hC.disjointOfDiff (C := C) (s := f a) (t := f a ∩ s) ⋯ ⋯, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
have refine_2 : LE.le (α := ℝ≥0∞)
  (∑' (i : ℕ), ((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (f i \ s))
  (∑' (a : ℕ), ∑ u ∈ hC.disjointOfDiff (C := C) (s := f a) (t := f a ∩ s) sorry sorry, (m : (a : Set α) → ℝ≥0∞) u)
Tactic: apply le_trans <| measure_iUnion_le (μ := OuterMeasure.ofFunction m addContent_empty)
      (fun i ↦ f i \ s)
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞)
    (∑' (i : ℕ),
      ((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (f i \ s))
    (∑' (a : ℕ), ∑ u ∈ hC.disjointOfDiff (C := C) (s := f a) (t := f a ∩ s) ⋯ ⋯, (m : (a : Set α) → ℝ≥0∞) u)
have refine_2 : ∀ (a : ℕ),
  LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (f a \ s) : ℝ≥0∞)
    (∑ u ∈ hC.disjointOfDiff (C := C) (s := f a) (t := f a ∩ s) sorry sorry, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
Tactic: apply ENNReal.tsum_le_tsum
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (hypothesis)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
i : ℕ
⊢ LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (f i \ s) : ℝ≥0∞)
    (∑ u ∈ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
have h_diff_eq_sUnion₁ : ∀ (i : ℕ), f i \ s = ⋃ i_1 ∈ (↑(A i) : Set (Set α)), i_1
Tactic: simp_rw [sUnion_eq_biUnion] at h_diff_eq_sUnion
Cleared hypotheses: h_diff_eq_sUnion
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
i : ℕ
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃ i_1 ∈ (↑(A i) : Set (Set α)), i_1
⊢ LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (f i \ s) : ℝ≥0∞)
    (∑ u ∈ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
have refine_2 : LE.le (α := ℝ≥0∞)
  (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞)
      (⋃ i_1 ∈ (↑(A i) : Set (Set α)), i_1) :
    ℝ≥0∞)
  (∑ u ∈ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) sorry sorry, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
Tactic: rw [h_diff_eq_sUnion]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (hypothesis)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
i : ℕ
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃ i_1 ∈ (↑(A i) : Set (Set α)), i_1
⊢ LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞)
        (⋃ i_1 ∈ (↑(A i) : Set (Set α)), i_1) :
      ℝ≥0∞)
    (∑ u ∈ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
have h6 : LE.le (α := ℝ≥0∞)
  (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (⋃ i_1 ∈ A i, id i_1) :
    ℝ≥0∞)
  (∑ i ∈ A i, ((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (id i) :
    ℝ≥0∞)
Tactic: obtain h6 := MeasureTheory.measure_biUnion_finset_le
      (μ := OuterMeasure.ofFunction m addContent_empty) (A i) id
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (hypothesis)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_m_eq :
  ∀ (i : ℕ),
    (m : (a : Set α) → ℝ≥0∞) (f i) =
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
i : ℕ
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃ i_1 ∈ (↑(A i) : Set (Set α)), i_1
h6 :
  LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞)
        (⋃ i_1 ∈ A i, id i_1) :
      ℝ≥0∞)
    (∑ i ∈ A i, ((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (id i) :
      ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞)
        (⋃ i_1 ∈ (↑(A i) : Set (Set α)), i_1) :
      ℝ≥0∞)
    (∑ u ∈ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
have h6₁ : LE.le (α := ℝ≥0∞)
  (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (⋃ i_1 ∈ A i, i_1) :
    ℝ≥0∞)
  (∑ x ∈ A i, ((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞)
Tactic: simp only [id_eq] at h6
Cleared hypotheses: h6
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
    ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i)
have have : ∀
  (h_m_eq :
    ∀ (i : ℕ),
      (m : (a : Set α) → ℝ≥0∞) (f i) =
        HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
          (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)),
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
    ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i)
Tactic: classical
  have h_m_eq i : m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u :=
    eq_add_disjointOfDiff_of_subset hC (hC.inter_mem (f i) (hf i) s hs) (hf i) inter_subset_left
  simp_rw [h_m_eq]
  rw [ENNReal.tsum_add]
  refine add_le_add ?_ ?_
  · refine iInf_le_of_le (fun i ↦ f i ∩ s) <| iInf_le_of_le ?_ le_rfl
    rw [← iUnion_inter]
    exact Set.inter_subset_inter_left _ hf_subset
  · apply le_trans <| (OuterMeasure.ofFunction m addContent_empty).mono
      <| (iUnion_diff s f) ▸ diff_subset_diff_left hf_subset
    simp only [OuterMeasure.measureOf_eq_coe, A]
    apply le_trans <| measure_iUnion_le (μ := OuterMeasure.ofFunction m addContent_empty)
      (fun i ↦ f i \ s)
    apply ENNReal.tsum_le_tsum
    intro i
    simp_rw [sUnion_eq_biUnion] at h_diff_eq_sUnion
    rw [h_diff_eq_sUnion]
    obtain h6 := MeasureTheory.measure_biUnion_finset_le
      (μ := OuterMeasure.ofFunction m addContent_empty) (A i) id
    simp only [id_eq] at h6
    exact le_trans h6 <| Finset.sum_le_sum <| fun b _ ↦ OuterMeasure.ofFunction_le b
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.OfAddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ (i : ℕ), f i
A : ℕ → Finset (Set α) := fun (i : ℕ) ↦ hC.disjointOfDiff (C := C) (s := f i) (t := f i ∩ s) ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
have have₁ have₂ :
  ∀
    (h_m_eq :
      ∀ (i : ℕ),
        (m : (a : Set α) → ℝ≥0∞) (f i) =
          HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
            (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
        (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
        (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
      ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
    ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i)
have have₃ : ∀
  (h_m_eq :
    ∀ (i : ℕ),
      (m : (a : Set α) → ℝ≥0∞) (f i) =
        HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
          (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)),
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t ∩ s) : ℝ≥0∞)
      (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) (t \ s) : ℝ≥0∞) ≤
    ∑' (i : ℕ),
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i ∩ s) : ℝ≥0∞)
        (∑ u ∈ A i, (m : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞)
Tactic: classical
  have h_m_eq i : m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u :=
    eq_add_disjointOfDiff_of_subset hC (hC.inter_mem (f i) (hf i) s hs) (hf i) inter_subset_left
  simp_rw [h_m_eq]
  rw [ENNReal.tsum_add]
  refine add_le_add ?_ ?_
  · refine iInf_le_of_le (fun i ↦ f i ∩ s) <| iInf_le_of_le ?_ le_rfl
    rw [← iUnion_inter]
    exact Set.inter_subset_inter_left _ hf_subset
  · apply le_trans <| (OuterMeasure.ofFunction m addContent_empty).mono
      <| (iUnion_diff s f) ▸ diff_subset_diff_left hf_subset
    simp only [OuterMeasure.measureOf_eq_coe, A]
    apply le_trans <| measure_iUnion_le (μ := OuterMeasure.ofFunction m addContent_empty)
      (fun i ↦ f i \ s)
    apply ENNReal.tsum_le_tsum
    intro i
    simp_rw [sUnion_eq_biUnion] at h_diff_eq_sUnion
    rw [h_diff_eq_sUnion]
    obtain h6 := MeasureTheory.measure_biUnion_finset_le
      (μ := OuterMeasure.ofFunction m addContent_empty) (A i) id
    simp only [id_eq] at h6
    exact le_trans h6 <| Finset.sum_le_sum <| fun b _ ↦ OuterMeasure.ofFunction_le b