awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ f j ≠ f j

Tactic:
rw [h_eq] at hst
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
hst₁ : f j ≠ f j
⊢ i = j

Tactic:
rw [h_eq] at hst
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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 j ≠ f j
h_eq : i = j
⊢ False

Tactic:
hst rfl

Premises used: rfl
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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 j ≠ f j
h_eq : i = j
⊢ False

Tactic:
exact hst rfl

Premises used: rfl
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ i ≠ j

Tactic:
have hij : i ≠ j := by intro h_eq; rw [h_eq] at hst; exact hst rfl
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
hij : i ≠ j
⊢ (Disjoint on id) (f i) (f j)

Tactic:
hf_disj hij
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
hij : i ≠ j
⊢ (Disjoint on id) (f i) (f j)

Tactic:
exact hf_disj hij
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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_3 : ∀ a ∈ I, f a ⊆ ⋃ (i : ℕ), f i
⊢ ∀ s ∈ Finset.image f I, s ⊆ ⋃ (i : ℕ), f i

Tactic:
simp only [Finset.mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]

Premises used: Finset.mem_image, and_imp, forall_exists_index, forall_apply_eq_imp_iff₂
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ ∀ a ∈ I, f a ⊆ ⋃ (i : ℕ), f i

Tactic:
fun i _ ↦ subset_iUnion _ i

Premises used: Set.subset_iUnion
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ ∀ a ∈ I, f a ⊆ ⋃ (i : ℕ), f i

Tactic:
exact fun i _ ↦ subset_iUnion _ i

Premises used: Set.subset_iUnion
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
refine_3 : ∀ s ∈ Finset.image f I, s ⊆ ⋃ (i : ℕ), f i
refine_2₁ : (↑(Finset.image f I) : Set (Set α)).PairwiseDisjoint id
refine_3₁ refine_3₂ : ∀ s ∈ Finset.image f I, s ⊆ ⋃ (i : ℕ), f i
⊢ 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:
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

Premises used: MeasureTheory.sum_addContent_le_of_subset, Finset.mem_image, and_imp, Set.mem_image, forall_exists_index, Finset.coe_image, HasSubset.Subset.trans, Finset.sum_image_of_disjoint, Set.subset_iUnion, Set.preimage_mono, SetLike.coe, Set.image_subset_iff, Pairwise.pairwiseDisjoint, forall_apply_eq_imp_iff₂, Set.subset_preimage_image, rfl
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd : m.IsSigmaSubadditive (C := C)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
⊢ Eq (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∞)

Tactic:
addContent_iUnion_eq_tsum_of_disjoint_of_addContent_iUnion_le hC
    (fun _ hf hf_Union _ ↦ m_subadd hf hf_Union) f hf hf_Union hf_disj

Premises used: MeasureTheory.addContent_iUnion_eq_tsum_of_disjoint_of_addContent_iUnion_le
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ ∃ (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⟩
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
simp :
  f ⋯.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2 -
      f ⋯.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
⊢ onIocAux f (Set.Ioc u v) = f v - f u

Tactic:
simp only [onIocAux, h', ↓reduceDIte]

Premises used: MeasureTheory.AddContent.onIocAux, reduceDIte
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
set :
  let u' := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).1;
  f ⋯.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2 - f u' = f v - f u
⊢ f ⋯.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2 -
      f ⋯.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:
set u' := h'.choose.1

Premises used: Prod.fst, Exists.choose
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
set :
  let v' := 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
⊢ f ⋯.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2).2 - f u' = f v - f u

Tactic:
set v' := h'.choose.2

Premises used: Exists.choose, Prod.snd
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ 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
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
inr : ∀ (huv : u < v), f v' - f u' = f v - f u
⊢ f v' - f u' = f v - f u

Tactic:
rcases h.eq_or_lt with rfl | huv

Premises used: LE.le.eq_or_lt, rfl
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
u : α
h : u ≤ u
h' : ∃ (p : α × α), p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2
u' : α := h'.choose (p := fun (p : α × α) ↦ p.1 ≤ p.2 ∧ Set.Ioc u u = Set.Ioc p.1 p.2).1
v' : α := 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:
grind [Set.Ioc_eq_empty_iff]

Premises used: Set.Ioc_eq_empty_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ u' ≤ v' ∧ u = u' ∧ v = v'

Tactic:
rw [Ioc_eq_Ioc_iff (Or.inl huv)] at hu'v'

Premises used: Or.inl, Set.Ioc_eq_Ioc_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
hu'v'₁ : u' ≤ v' ∧ u = u' ∧ v = v'
⊢ u < v

Tactic:
rw [Ioc_eq_Ioc_iff (Or.inl huv)] at hu'v'

Premises used: Or.inl, Set.Ioc_eq_Ioc_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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' ∧ u = u' ∧ v = v'
huv : u < v
⊢ f v' - f u' = f v - f u

Tactic:
grind
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
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
⊢ onIocAux f ∅ = 0

Tactic:
rw [onIocAux, dite_eq_right_iff]

Premises used: MeasureTheory.AddContent.onIocAux, dite_eq_right_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
inst : LinearOrder α
G : Type u_3
inst_1 : AddCommGroup G
f : α → G
⊢ ∀ (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:
grind [Set.Ioc_eq_empty_iff]

Premises used: Set.Ioc_eq_empty_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
rw₁ rw₂ rw₃ :
  ∀ (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
⊢ onIocAux f ∅ = 0

Tactic:
classical
  rw [onIocAux, dite_eq_right_iff]
  grind [Set.Ioc_eq_empty_iff]

Premises used: MeasureTheory.AddContent.onIocAux, dite_eq_right_iff, Set.Ioc_eq_empty_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ I = ∅

Tactic:
grind
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
⊢ I = ∅

Tactic:
have : I = ∅ := by grind
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
this : I = ∅
⊢ onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u

Tactic:
simp [this, onIocAux_empty f]

Premises used: MeasureTheory.AddContent.onIocAux_empty
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
n : ℕ
ih :
  ∀ (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 = n), onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
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 = n + 1
succ :
  ∀ (u v : α) (huv : u ≤ v) (h'uv : ⋃₀ (↑I : Set (Set α)) = Set.Ioc u v),
    onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
⊢ onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u

Tactic:
rcases h''I with ⟨u, v, huv, h'uv⟩
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
n : ℕ
ih :
  ∀ (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 = n), onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
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
hn : #I = n + 1
u v : α
huv : u ≤ v
h'uv : ⋃₀ (↑I : Set (Set α)) = Set.Ioc u v
succ :
  ∀ (huv : u ≤ u) (h'uv : ⋃₀ (↑I : Set (Set α)) = Set.Ioc u u),
    onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
succ₁ : ∀ (huv : u < v), onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
⊢ onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u

Tactic:
rcases huv.eq_or_lt with rfl | huv

Premises used: LE.le.eq_or_lt, rfl
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
n : ℕ
ih :
  ∀ (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 = n), onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
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
hn : #I = n + 1
u : α
huv : u ≤ u
h'uv : ⋃₀ (↑I : Set (Set α)) = Set.Ioc u u
⊢ onIocAux f (Set.Ioc u u) = ∑ u ∈ I, 0

Tactic:
simp [onIocAux_empty f]

Premises used: MeasureTheory.AddContent.onIocAux_empty
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
n : ℕ
ih :
  ∀ (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 = n), onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
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
hn : #I = n + 1
u : α
huv : u ≤ u
h'uv : ⋃₀ (↑I : Set (Set α)) = Set.Ioc u u
⊢ onIocAux f (Set.Ioc u u) = ∑ u ∈ I, 0

Tactic:
have : onIocAux f (Set.Ioc u u) = ∑ u ∈ I, 0 := by simp [onIocAux_empty f]
-- Module: Mathlib.MeasureTheory.Measure.AddContent
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
n : ℕ
ih :
  ∀ (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 = n), onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u
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
hn : #I = n + 1
u : α
huv : u ≤ u
h'uv : ⋃₀ (↑I : Set (Set α)) = Set.Ioc u u
this : onIocAux f (Set.Ioc u u) = ∑ u ∈ I, 0
succ : Eq.{u_3 + 1} (α := G) (∑ u ∈ I, 0) (∑ u ∈ I, onIocAux f u)
⊢ onIocAux f (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, onIocAux f u

Tactic:
rw [h'uv, this]