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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
⊢ Eq (α := ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
have refine : LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
  (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
Tactic: refine le_antisymm (OuterMeasure.ofFunction_le 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
⊢ LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
have rw : LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
  (⨅ (t : ℕ → Set α),
      ⨅ (_ : ∀ (i : ℕ), t i ∈ C), ⨅ (_ : s ⊆ ⋃ (i : ℕ), t i), ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (t i) :
    ℝ≥0∞)
Tactic: rw [OuterMeasure.ofFunction_eq_iInf_mem _ _ m_top]
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
⊢ LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    (⨅ (t : ℕ → Set α),
        ⨅ (_ : ∀ (i : ℕ), t i ∈ C), ⨅ (_ : s ⊆ ⋃ (i : ℕ), t i), ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (t i) :
      ℝ≥0∞)
have refine : ∀ (f : ℕ → Set α) (hf : ∀ (i : ℕ), f i ∈ C) (hs_subset : s ⊆ ⋃ (i : ℕ), f i),
  LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
Tactic: refine le_iInf fun f ↦ le_iInf fun hf ↦ le_iInf fun hs_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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
have rw : Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
Tactic: rw [inter_eq_self_of_subset_left hs_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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
have rw : Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
  ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
Tactic: rw [inter_iUnion]
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
⊢ LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
    (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (s ∩ f i) : ℝ≥0∞)
have refine : ⋃ (i : ℕ), s ∩ f i ∈ C
Tactic: refine m_sigma_subadd (fun i ↦ hC.inter_mem _ hs _ (hf i)) ?_
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
⊢ ⋃ (i : ℕ), s ∩ f i ∈ C
have rwa : s ∩ ⋃ (i : ℕ), f i ∈ C
Tactic: rwa [← inter_iUnion, inter_eq_self_of_subset_left hs_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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
⊢ LE.le (α := ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (s ∩ f i)) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i))
have refine : ∀ (i : ℕ), LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
Tactic: refine ENNReal.summable.tsum_le_tsum (fun i ↦ ?_) ENNReal.summable
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
⊢ LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
have calc : Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
Tactic: calc m s = m (s ∩ ⋃ i, f i) := by rw [inter_eq_self_of_subset_left hs_subset]
    _ = m (⋃ i, s ∩ f i) := by rw [inter_iUnion]
    _ ≤ ∑' i, m (s ∩ f i) := by
      refine m_sigma_subadd (fun i ↦ hC.inter_mem _ hs _ (hf i)) ?_
      rwa [← inter_iUnion, inter_eq_self_of_subset_left hs_subset]
    _ ≤ ∑' i, m (f i) := by
      refine ENNReal.summable.tsum_le_tsum (fun i ↦ ?_) ENNReal.summable
      exact addContent_mono hC (hC.inter_mem _ hs _ (hf i)) (hf i) Set.inter_subset_right
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
calc : Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
have calc₁ : Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
  ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
Tactic: calc m s = m (s ∩ ⋃ i, f i) := by rw [inter_eq_self_of_subset_left hs_subset]
    _ = m (⋃ i, s ∩ f i) := by rw [inter_iUnion]
    _ ≤ ∑' i, m (s ∩ f i) := by
      refine m_sigma_subadd (fun i ↦ hC.inter_mem _ hs _ (hf i)) ?_
      rwa [← inter_iUnion, inter_eq_self_of_subset_left hs_subset]
    _ ≤ ∑' i, m (f i) := by
      refine ENNReal.summable.tsum_le_tsum (fun i ↦ ?_) ENNReal.summable
      exact addContent_mono hC (hC.inter_mem _ hs _ (hf i)) (hf i) Set.inter_subset_right
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
calc : Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
calc₁ :
  Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
have calc₂ : LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
  (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (s ∩ f i) : ℝ≥0∞)
Tactic: calc m s = m (s ∩ ⋃ i, f i) := by rw [inter_eq_self_of_subset_left hs_subset]
    _ = m (⋃ i, s ∩ f i) := by rw [inter_iUnion]
    _ ≤ ∑' i, m (s ∩ f i) := by
      refine m_sigma_subadd (fun i ↦ hC.inter_mem _ hs _ (hf i)) ?_
      rwa [← inter_iUnion, inter_eq_self_of_subset_left hs_subset]
    _ ≤ ∑' i, m (f i) := by
      refine ENNReal.summable.tsum_le_tsum (fun i ↦ ?_) ENNReal.summable
      exact addContent_mono hC (hC.inter_mem _ hs _ (hf i)) (hf i) Set.inter_subset_right
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
m_top : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ (i : ℕ), f i
calc : Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
calc₁ :
  Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (s ∩ ⋃ (i : ℕ), f i) : ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
calc₂ :
  LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), s ∩ f i) : ℝ≥0∞)
    (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (s ∩ f i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
have calc₃ : LE.le (α := ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (s ∩ f i)) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i))
Tactic: calc m s = m (s ∩ ⋃ i, f i) := by rw [inter_eq_self_of_subset_left hs_subset]
    _ = m (⋃ i, s ∩ f i) := by rw [inter_iUnion]
    _ ≤ ∑' i, m (s ∩ f i) := by
      refine m_sigma_subadd (fun i ↦ hC.inter_mem _ hs _ (hf i)) ?_
      rwa [← inter_iUnion, inter_eq_self_of_subset_left hs_subset]
    _ ≤ ∑' i, m (f i) := by
      refine ENNReal.summable.tsum_le_tsum (fun i ↦ ?_) ENNReal.summable
      exact addContent_mono hC (hC.inter_mem _ hs _ (hf i)) (hf i) Set.inter_subset_right
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
this :
  Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) ⋯ ⋯ :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) ⋯ ⋯ :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
have rwa : ∀
  (this :
    Eq (α := ℝ≥0∞)
      ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) sorry sorry :
            (a : Set α) → ℝ≥0∞)
          s :
        ℝ≥0∞)
      ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)),
  Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) sorry sorry :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
Tactic: rwa [m.extend_eq hC hs] at this
Cleared hypotheses: this
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
⊢ Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) ⋯ ⋯ :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    ((m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
have suffices : Eq (α := ℝ≥0∞)
  ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) sorry sorry :
        (a : Set α) → ℝ≥0∞)
      s :
    ℝ≥0∞)
  ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
Tactic: suffices inducedOuterMeasure (fun x _ ↦ m x) hC.empty_mem addContent_empty s = m.extend hC s by
    rwa [m.extend_eq hC hs] at this
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
⊢ Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) ⋯ ⋯ :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
have refine_1 : Eq (α := ℝ≥0∞)
  ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) sorry sorry :
        (a : Set α) → ℝ≥0∞)
      s :
    ℝ≥0∞)
  (((OuterMeasure.ofFunction (⇑(AddContent.extend (C := C) hC m) : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) :
        (a : Set α) → ℝ≥0∞)
      s :
    ℝ≥0∞)
Tactic: refine Eq.trans ?_ ((m.extend hC).ofFunction_eq hC ?_ ?_ hs)
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
refine_1 :
  Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) ⋯ ⋯ :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑(AddContent.extend (C := C) hC m) : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) ⋯ ⋯ :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
have refine_2 : (AddContent.extend (C := C) hC m).IsSigmaSubadditive (C := C)
Tactic: refine Eq.trans ?_ ((m.extend hC).ofFunction_eq hC ?_ ?_ hs)
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
refine_1 :
  Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) ⋯ ⋯ :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    (((OuterMeasure.ofFunction (⇑(AddContent.extend (C := C) hC m) : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
refine_2 : (AddContent.extend (C := C) hC m).IsSigmaSubadditive (C := C)
⊢ Eq (α := ℝ≥0∞)
    ((inducedOuterMeasure (P := Membership.mem C) (fun (x : Set α) (x_1 : x ∈ C) ↦ (m : (a : Set α) → ℝ≥0∞) x) ⋯ ⋯ :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
    ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
have refine_3 : ∀ s ∉ C, Eq (α := ℝ≥0∞) ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: refine Eq.trans ?_ ((m.extend hC).ofFunction_eq hC ?_ ?_ hs)
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_mem : ⋃ (i : ℕ), f i ∈ C
⊢ LE.le (α := ℝ≥0∞) ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
    (∑' (i : ℕ), (AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
have refine_2 : LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
  (∑' (i : ℕ), (AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
Tactic: rw [m.extend_eq hC hf_mem]
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_mem : ⋃ (i : ℕ), f i ∈ C
⊢ LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
    (∑' (i : ℕ), (AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
have refine_2 : Eq (α := ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i))
  (∑' (i : ℕ), (AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) (f i))
Tactic: refine (m_sigma_subadd hf hf_mem).trans_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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_mem : ⋃ (i : ℕ), f i ∈ C
⊢ Eq (α := ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i))
    (∑' (i : ℕ), (AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) (f i))
have refine_2 : ∀ (i : ℕ),
  Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
    ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
Tactic: congr with i
-- 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_sigma_subadd : m.IsSigmaSubadditive (C := C)
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_mem : ⋃ (i : ℕ), f i ∈ C
i : ℕ
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
    ((AddContent.extend (C := C) hC m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
have refine_2 : Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
Tactic: rw [m.extend_eq hC (hf i)]
-- 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
⊢ OuterMeasure.IsCaratheodory (OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯) s
have rw : ∀ (t : Set α),
  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∞) ≤
    ((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) sorry : OuterMeasure α) : (a : Set α) → ℝ≥0∞) t
Tactic: rw [OuterMeasure.isCaratheodory_iff_le']
-- 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
| (((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have rw : ⨅ (t_1 : ℕ → Set α),
  ⨅ (_ : ∀ (i : ℕ), t_1 i ∈ C), ⨅ (_ : t ⊆ ⋃ (i : ℕ), t_1 i), ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (t_1 i)
Tactic: rw [OuterMeasure.ofFunction_eq_iInf_mem _ _ m_top]
-- 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
⊢ 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∞) ≤
    ((OuterMeasure.ofFunction (⇑m : (a : Set α) → ℝ≥0∞) ⋯ : OuterMeasure α) : (a : Set α) → ℝ≥0∞) t
have conv_rhs : 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∞) ≤
  ⨅ (t_1 : ℕ → Set α),
    ⨅ (_ : ∀ (i : ℕ), t_1 i ∈ C), ⨅ (_ : t ⊆ ⋃ (i : ℕ), t_1 i), ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (t_1 i)
Tactic: conv_rhs => rw [OuterMeasure.ofFunction_eq_iInf_mem _ _ m_top]
-- 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
⊢ 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∞) ≤
    ⨅ (t_1 : ℕ → Set α),
      ⨅ (_ : ∀ (i : ℕ), t_1 i ∈ C), ⨅ (_ : t ⊆ ⋃ (i : ℕ), t_1 i), ∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (t_1 i)
have refine : ∀ (f : ℕ → Set α) (hf : ∀ (i : ℕ), f i ∈ C) (hf_subset : t ⊆ ⋃ (i : ℕ), f i),
  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: refine le_iInf fun f ↦ le_iInf fun hf ↦ le_iInf fun hf_subset ↦ ?_
-- 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) ⋯ ⋯
⊢ 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 h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ (↑(A i) : Set (Set α))
Tactic: have h_diff_eq_sUnion i : f i \ s = ⋃₀ A i := by simp [A, IsSetSemiring.sUnion_disjointOfDiff]
-- 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))
⊢ 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 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∞)
Tactic: 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