awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
h_ss : (↑∅ : Set (Set α)) ⊆ C
h_dis : (↑∅ : Set (Set α)).PairwiseDisjoint id
h_mem : ⋃₀ (↑∅ : Set (Set α)) ∈ C
⊢ m (⋃₀ (↑∅ : Set (Set α))) = ∑ u ∈ ∅, m u

Tactic:
simp only [Finset.coe_empty, Set.sUnion_empty, Finset.sum_empty, m_empty]

Premises used: Finset.coe_empty, Finset.sum_empty, Set.sUnion_empty
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : (↑(Insert.insert s I) : Set (Set α)) ⊆ C
h_dis : (↑(Insert.insert s I) : Set (Set α)).PairwiseDisjoint id
h_mem : ⋃₀ (↑(Insert.insert s I) : Set (Set α)) ∈ C
insert :
  ∀ (h_ss : Insert.insert s (↑I : Set (Set α)) ⊆ C) (h_dis : (Insert.insert s (↑I : Set (Set α))).PairwiseDisjoint id)
    (h_mem : ⋃₀ Insert.insert s (↑I : Set (Set α)) ∈ C),
    m (⋃₀ Insert.insert s (↑I : Set (Set α))) = ∑ u ∈ Insert.insert s I, m u
⊢ m (⋃₀ (↑(Insert.insert s I) : Set (Set α))) = ∑ u ∈ Insert.insert s I, m u

Tactic:
rw [Finset.coe_insert] at *

Premises used: Finset.coe_insert
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : insert s (↑I : Set (Set α)) ⊆ C
h_dis : (insert s (↑I : Set (Set α))).PairwiseDisjoint id
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
⊢ s ∈ C ∧ (↑I : Set (Set α)) ⊆ C

Tactic:
rw [Set.insert_subset_iff] at h_ss

Premises used: Set.insert_subset_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : insert s (↑I : Set (Set α)) ⊆ C
h_dis : (insert s (↑I : Set (Set α))).PairwiseDisjoint id
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
h_ss₁ : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
⊢ (insert s (↑I : Set (Set α))).PairwiseDisjoint id

Tactic:
rw [Set.insert_subset_iff] at h_ss

Premises used: Set.insert_subset_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : insert s (↑I : Set (Set α)) ⊆ C
h_dis : (insert s (↑I : Set (Set α))).PairwiseDisjoint id
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
h_ss₁ : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis₁ : (insert s (↑I : Set (Set α))).PairwiseDisjoint id
⊢ ⋃₀ insert s (↑I : Set (Set α)) ∈ C

Tactic:
rw [Set.insert_subset_iff] at h_ss

Premises used: Set.insert_subset_iff
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (Insert.insert s (↑I : Set (Set α))).PairwiseDisjoint id
h_mem : ⋃₀ Insert.insert s (↑I : Set (Set α)) ∈ C
insert :
  ∀ (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id ∧ ∀ j ∈ (↑I : Set (Set α)), Disjoint (id s) (id j))
    (h_mem : ⋃₀ Insert.insert s (↑I : Set (Set α)) ∈ C),
    m (⋃₀ Insert.insert s (↑I : Set (Set α))) = ∑ u ∈ Insert.insert s I, m u
insert₁ : s ∉ (↑I : Set (Set α))
⊢ m (⋃₀ Insert.insert s (↑I : Set (Set α))) = ∑ u ∈ Insert.insert s I, m u

Tactic:
rw [Set.pairwiseDisjoint_insert_of_notMem] at h_dis

Premises used: Set.pairwiseDisjoint_insert_of_notMem
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (insert s (↑I : Set (Set α))).PairwiseDisjoint id
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
⊢ s ∉ (↑I : Set (Set α))

Tactic:
hsI
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (insert s (↑I : Set (Set α))).PairwiseDisjoint id
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
⊢ s ∉ (↑I : Set (Set α))

Tactic:
exact hsI
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id ∧ ∀ j ∈ (↑I : Set (Set α)), Disjoint (id s) (id j)
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
rw : ⋃ i ∈ (↑I : Set (Set α)), i ∈ C
⊢ ⋃₀ (↑I : Set (Set α)) ∈ C

Tactic:
rw [Set.sUnion_eq_biUnion]

Premises used: Set.sUnion_eq_biUnion
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id ∧ ∀ j ∈ (↑I : Set (Set α)), Disjoint (id s) (id j)
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
hs : ∀ n ∈ I, n ∈ C
⊢ ⋃ i ∈ (↑I : Set (Set α)), i ∈ C

Tactic:
apply hC.biUnion_mem

Premises used: MeasureTheory.IsSetRing.biUnion_mem
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id ∧ ∀ j ∈ (↑I : Set (Set α)), Disjoint (id s) (id j)
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
n : Set α
hn : n ∈ I
⊢ n ∈ C

Tactic:
h_ss.2 hn

Premises used: And.right
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id ∧ ∀ j ∈ (↑I : Set (Set α)), Disjoint (id s) (id j)
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
n : Set α
hn : n ∈ I
⊢ n ∈ C

Tactic:
exact h_ss.2 hn

Premises used: And.right
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id ∧ ∀ j ∈ (↑I : Set (Set α)), Disjoint (id s) (id j)
h_mem : ⋃₀ insert s (↑I : Set (Set α)) ∈ C
⊢ ⋃₀ (↑I : Set (Set α)) ∈ C

Tactic:
have h_sUnion_mem : ⋃₀ ↑I ∈ C := by
        rw [Set.sUnion_eq_biUnion]
        apply hC.biUnion_mem
        intro n hn
        exact h_ss.2 hn
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id ∧ ∀ j ∈ (↑I : Set (Set α)), Disjoint (id s) (id j)
h_mem : ⋃₀ Insert.insert s (↑I : Set (Set α)) ∈ C
h_sUnion_mem insert : ⋃₀ (↑I : Set (Set α)) ∈ C
⊢ m (⋃₀ Insert.insert s (↑I : Set (Set α))) = ∑ u ∈ Insert.insert s I, m u

Tactic:
rw [Set.sUnion_insert, m_add h_ss.1 h_sUnion_mem (Set.disjoint_sUnion_right.mpr h_dis.2),
        Finset.sum_insert hsI, h h_ss.2 h_dis.1]

Premises used: Iff.mpr, Set.sUnion_insert, And.right, And.left, Set.disjoint_sUnion_right, Finset.sum_insert
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s_1 t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
s : Set α
I : Finset (Set α)
hsI : s ∉ I
h :
  ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u
h_ss : s ∈ C ∧ (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id ∧ ∀ j ∈ (↑I : Set (Set α)), Disjoint (id s) (id j)
h_mem : ⋃₀ Insert.insert s (↑I : Set (Set α)) ∈ C
h_sUnion_mem : ⋃₀ (↑I : Set (Set α)) ∈ C
insert insert₁ insert₂ insert₃ :
  ∀ (h_mem : s ∪ ⋃₀ (↑I : Set (Set α)) ∈ C) (h_sUnion_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), ⋃₀ (↑I : Set (Set α)) ∈ C
⊢ ⋃₀ (↑I : Set (Set α)) ∈ C

Tactic:
rwa [Set.sUnion_insert] at h_mem

Premises used: Set.sUnion_insert
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s t : Set α
I_1 : Finset (Set α)
G : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
I : Finset (Set α)
h_ss : (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id
h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C
empty :
  ∀ (h_ss : (↑∅ : Set (Set α)) ⊆ C) (h_dis : (↑∅ : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑∅ : Set (Set α)) ∈ C), m (⋃₀ (↑∅ : Set (Set α))) = ∑ u ∈ ∅, m u
insert :
  ∀ (s : Set α) (I : Finset (Set α)) (hsI : s ∉ I)
    (h :
      ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
        (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u)
    (h_ss : (↑(Insert.insert s I) : Set (Set α)) ⊆ C) (h_dis : (↑(Insert.insert s I) : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑(Insert.insert s I) : Set (Set α)) ∈ C),
    m (⋃₀ (↑(Insert.insert s I) : Set (Set α))) = ∑ u ∈ Insert.insert s I, m u
⊢ m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u

Tactic:
induction I using Finset.induction with
    | empty => simp only [Finset.coe_empty, Set.sUnion_empty, Finset.sum_empty, m_empty]
    | insert s I hsI h =>
      rw [Finset.coe_insert] at *
      rw [Set.insert_subset_iff] at h_ss
      rw [Set.pairwiseDisjoint_insert_of_notMem] at h_dis
      swap; · exact hsI
      have h_sUnion_mem : ⋃₀ ↑I ∈ C := by
        rw [Set.sUnion_eq_biUnion]
        apply hC.biUnion_mem
        intro n hn
        exact h_ss.2 hn
      rw [Set.sUnion_insert, m_add h_ss.1 h_sUnion_mem (Set.disjoint_sUnion_right.mpr h_dis.2),
        Finset.sum_insert hsI, h h_ss.2 h_dis.1]
      rwa [Set.sUnion_insert] at h_mem

Premises used: Finset.coe_empty, Set.sUnion_insert, Set.pairwiseDisjoint_insert_of_notMem, Insert.insert, Set.insert_subset_iff, Finset.sum_empty, Finset.coe_insert, Set.sUnion_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 : Type u_2
inst : AddCommMonoid G
m_1 m' : AddContent G C
m : Set α → G
hC : IsSetRing C
m_empty : m ∅ = 0
m_add : ∀ {s t : Set α} (_hs : s ∈ C) (_ht : t ∈ C), Disjoint s t → m (s ∪ t) = m s + m t
I : Finset (Set α)
h_ss : (↑I : Set (Set α)) ⊆ C
h_dis : (↑I : Set (Set α)).PairwiseDisjoint id
h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C
empty :
  ∀ (h_ss : (↑∅ : Set (Set α)) ⊆ C) (h_dis : (↑∅ : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑∅ : Set (Set α)) ∈ C), m (⋃₀ (↑∅ : Set (Set α))) = ∑ u ∈ ∅, m u
insert :
  ∀ (s : Set α) (I : Finset (Set α)) (hsI : s ∉ I)
    (h :
      ∀ (h_ss : (↑I : Set (Set α)) ⊆ C) (h_dis : (↑I : Set (Set α)).PairwiseDisjoint id)
        (h_mem : ⋃₀ (↑I : Set (Set α)) ∈ C), m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u)
    (h_ss : (↑(Insert.insert s I) : Set (Set α)) ⊆ C) (h_dis : (↑(Insert.insert s I) : Set (Set α)).PairwiseDisjoint id)
    (h_mem : ⋃₀ (↑(Insert.insert s I) : Set (Set α)) ∈ C),
    m (⋃₀ (↑(Insert.insert s I) : Set (Set α))) = ∑ u ∈ Insert.insert s I, m u
⊢ m (⋃₀ (↑I : Set (Set α))) = ∑ u ∈ I, m u

Tactic:
classical
    induction I using Finset.induction with
    | empty => simp only [Finset.coe_empty, Set.sUnion_empty, Finset.sum_empty, m_empty]
    | insert s I hsI h =>
      rw [Finset.coe_insert] at *
      rw [Set.insert_subset_iff] at h_ss
      rw [Set.pairwiseDisjoint_insert_of_notMem] at h_dis
      swap; · exact hsI
      have h_sUnion_mem : ⋃₀ ↑I ∈ C := by
        rw [Set.sUnion_eq_biUnion]
        apply hC.biUnion_mem
        intro n hn
        exact h_ss.2 hn
      rw [Set.sUnion_insert, m_add h_ss.1 h_sUnion_mem (Set.disjoint_sUnion_right.mpr h_dis.2),
        Finset.sum_insert hsI, h h_ss.2 h_dis.1]
      rwa [Set.sUnion_insert] at h_mem

Premises used: Finset.coe_empty, Set.sUnion_insert, Set.pairwiseDisjoint_insert_of_notMem, Insert.insert, Set.insert_subset_iff, Finset.sum_empty, Finset.coe_insert, Set.sUnion_empty
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
hC : IsSetRing C
hs : s ∈ C
ht : t ∈ C
rw :
  HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) s : G) ((m : (a : Set α) → G) (t \ s) : G) ≤
    HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) s : G) ((m : (a : Set α) → G) t : G)
rw₁ : Disjoint s (t \ s)
⊢ (m : (a : Set α) → G) (s ∪ t) ≤
    HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) s : G) ((m : (a : Set α) → G) t : G)

Tactic:
rw [← union_diff_self, addContent_union hC hs (hC.diff_mem ht hs)]

Premises used: MeasureTheory.addContent_union, MeasureTheory.IsSetRing.diff_mem, Set.union_diff_self
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
hC : IsSetRing C
hs : s ∈ C
ht : t ∈ C
⊢ HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) s : G) ((m : (a : Set α) → G) (t \ s) : G) ≤
    HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) s : G) ((m : (a : Set α) → G) t : G)

Tactic:
add_le_add_right (addContent_mono hC.isSetSemiring (hC.diff_mem ht hs) ht diff_subset) _

Premises used: Set.diff_subset, add_le_add_right, MeasureTheory.addContent_mono, MeasureTheory.IsSetRing.isSetSemiring, MeasureTheory.IsSetRing.diff_mem
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
hC : IsSetRing C
hs : s ∈ C
ht : t ∈ C
⊢ HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) s : G) ((m : (a : Set α) → G) (t \ s) : G) ≤
    HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) s : G) ((m : (a : Set α) → G) t : G)

Tactic:
exact add_le_add_right (addContent_mono hC.isSetSemiring (hC.diff_mem ht hs) ht diff_subset) _

Premises used: Set.diff_subset, add_le_add_right, MeasureTheory.addContent_mono, MeasureTheory.IsSetRing.isSetSemiring, MeasureTheory.IsSetRing.diff_mem
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
hC : IsSetRing C
hs : s ∈ C
ht : t ∈ C
rw : s ∩ (t \ s) = ∅
rw₁ rw₂ : Eq.{u_1 + 1} (α := Set α) ∅ ∅
⊢ Disjoint s (t \ s)

Tactic:
rw [Set.disjoint_iff_inter_eq_empty, inter_diff_self]

Premises used: Set.inter_diff_self, Set.disjoint_iff_inter_eq_empty
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
ι : Type u_3
hC : IsSetRing C
s : ι → Set α
hs : ∀ (n : ι), Membership.mem (γ := Finset ι) ∅ n → s n ∈ C
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃ (i : ι), ⋃ (_ : Membership.mem (γ := Finset ι) ∅ i), s i) : G)
    (∑ i ∈ ∅, (m : (a : Set α) → G) (s i) : G)

Tactic:
simp
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
ι : Type u_3
hC : IsSetRing C
s : ι → Set α
i : ι
S : Finset ι
hiS : i ∉ S
h :
  ∀ (hs : ∀ n ∈ S, s n ∈ C),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃ i ∈ S, s i) : G) (∑ i ∈ S, (m : (a : Set α) → G) (s i) : G)
hs : ∀ n ∈ Insert.insert i S, s n ∈ C
insert :
  (m : (a : Set α) → G) (⋃ i_1 ∈ Insert.insert i S, s i_1) ≤
    HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) (s i) : G) (∑ x ∈ S, (m : (a : Set α) → G) (s x) : G)
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃ i_1 ∈ Insert.insert i S, s i_1) : G)
    (∑ i ∈ Insert.insert i S, (m : (a : Set α) → G) (s i) : G)

Tactic:
rw [Finset.sum_insert hiS]

Premises used: Finset.sum_insert
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
ι : Type u_3
hC : IsSetRing C
s : ι → Set α
i : ι
S : Finset ι
hiS : i ∉ S
h :
  ∀ (hs : ∀ n ∈ S, s n ∈ C),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃ i ∈ S, s i) : G) (∑ i ∈ S, (m : (a : Set α) → G) (s i) : G)
hs : ∀ n ∈ Insert.insert i S, s n ∈ C
insert :
  (m : (a : Set α) → G) (s i ∪ ⋃ x ∈ (↑S : Set ι), s x) ≤
    HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) (s i) : G) (∑ x ∈ S, (m : (a : Set α) → G) (s x) : G)
⊢ (m : (a : Set α) → G) (⋃ i_1 ∈ Insert.insert i S, s i_1) ≤
    HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) (s i) : G) (∑ x ∈ S, (m : (a : Set α) → G) (s x) : G)

Tactic:
simp_rw [← Finset.mem_coe, Finset.coe_insert, Set.biUnion_insert]

Premises used: Set.biUnion_insert, Finset.coe_insert, Finset.mem_coe
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
ι : Type u_3
hC : IsSetRing C
s : ι → Set α
i : ι
S : Finset ι
hiS : i ∉ S
h :
  ∀ (hs : ∀ n ∈ S, s n ∈ C),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃ i ∈ S, s i) : G) (∑ i ∈ S, (m : (a : Set α) → G) (s i) : G)
hs : ∀ n ∈ insert i S, s n ∈ C
⊢ s i ∈ C ∧ ∀ a ∈ S, s a ∈ C

Tactic:
simp only [Finset.mem_insert, forall_eq_or_imp] at hs

Premises used: Finset.mem_insert, forall_eq_or_imp
-- Module: Mathlib.MeasureTheory.Measure.AddContent
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
m : AddContent G C
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
ι : Type u_3
hC : IsSetRing C
s : ι → Set α
i : ι
S : Finset ι
hiS : i ∉ S
h :
  ∀ (hs : ∀ n ∈ S, s n ∈ C),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃ i ∈ S, s i) : G) (∑ i ∈ S, (m : (a : Set α) → G) (s i) : G)
hs : s i ∈ C ∧ ∀ a ∈ S, s a ∈ C
⊢ (m : (a : Set α) → G) (s i ∪ ⋃ x ∈ (↑S : Set ι), s x) ≤
    HAdd.hAdd (α := G) (β := G) ((m : (a : Set α) → G) (s i) : G) (∑ x ∈ S, (m : (a : Set α) → G) (s x) : G)

Tactic:
(addContent_union_le hC hs.1 (hC.biUnion_mem S hs.2)).trans ?_

Premises used: And.right, And.left, MeasureTheory.IsSetRing.biUnion_mem, LE.le.trans, MeasureTheory.addContent_union_le