awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
h : (↑(hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯) : Set (Set α)).PairwiseDisjoint id
⊢ (↑(K1 s) : Set (Set α)).PairwiseDisjoint id

Tactic:
rw [hK1s]
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
⊢ (↑(hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯) : Set (Set α)).PairwiseDisjoint id

Tactic:
hC.pairwiseDisjoint_disjointOfDiffUnion h1.1 h1.2

Premises used: And.right, And.left, MeasureTheory.IsSetSemiring.pairwiseDisjoint_disjointOfDiffUnion
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
⊢ (↑(hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯) : Set (Set α)).PairwiseDisjoint id

Tactic:
exact hC.pairwiseDisjoint_disjointOfDiffUnion h1.1 h1.2

Premises used: And.right, And.left, MeasureTheory.IsSetSemiring.pairwiseDisjoint_disjointOfDiffUnion
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
⊢ PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) id

Tactic:
simpa [ht2]
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
h : ∀ ⦃i : Set α⦄, i ∈ K1 s → ∀ ⦃j : Set α⦄, ∀ x ∈ J, j ∈ K1 x → ¬i = j → Disjoint i j
⊢ ∀ ⦃i : Set α⦄,
    i ∈ (↑(K1 s) : Set (Set α)) →
      ∀ ⦃j : Set α⦄,
        Membership.mem (γ := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) j → i ≠ j → Disjoint (id i) (id j)

Tactic:
simp only [mem_coe, mem_iUnion, exists_prop, ne_eq, id_eq, forall_exists_index, and_imp]

Premises used: and_imp, Set.mem_iUnion, forall_exists_index, exists_prop, ne_eq, Finset.mem_coe, id_eq
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
i : Set α
hi : i ∈ K1 s
j x : Set α
hx : x ∈ J
h3 : j ∈ K1 x
h4 : ¬i = j
⊢ i ⊆ s \ ⋃₀ (↑J : Set (Set α))

Tactic:
hC.subset_of_diffUnion_disjointOfDiffUnion h1.1 h1.2 _
          (hK1s ▸ hi)

Premises used: Eq.rec, And.right, And.left, MeasureTheory.IsSetSemiring.subset_of_diffUnion_disjointOfDiffUnion
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
i : Set α
hi : i ∈ K1 s
j x : Set α
hx : x ∈ J
h3 : j ∈ K1 x
h4 : ¬i = j
⊢ i ⊆ s \ ⋃₀ (↑J : Set (Set α))

Tactic:
obtain ki : i ⊆ s \ ⋃₀ J := hC.subset_of_diffUnion_disjointOfDiffUnion h1.1 h1.2 _
          (hK1s ▸ hi)

Premises used: And.right, And.left, MeasureTheory.IsSetSemiring.subset_of_diffUnion_disjointOfDiffUnion
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
i : Set α
hi : i ∈ K1 s
j x : Set α
hx : x ∈ J
h3 : j ∈ K1 x
h4 : ¬i = j
ki : i ⊆ s \ ⋃₀ (↑J : Set (Set α))
⊢ j ⊆ x

Tactic:
subset_trans (subset_sUnion_of_mem (ht1' x hx ▸ h3)) (hK3 x hx)

Premises used: subset_trans, Set.subset_sUnion_of_mem
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
i : Set α
hi : i ∈ K1 s
j x : Set α
hx : x ∈ J
h3 : j ∈ K1 x
h4 : ¬i = j
ki : i ⊆ s \ ⋃₀ (↑J : Set (Set α))
⊢ j ⊆ x

Tactic:
obtain hx2 : j ⊆ x := subset_trans (subset_sUnion_of_mem (ht1' x hx ▸ h3)) (hK3 x hx)

Premises used: subset_trans, Set.subset_sUnion_of_mem
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
i : Set α
hi : i ∈ K1 s
j x : Set α
hx : x ∈ J
h3 : j ∈ K1 x
h4 : ¬i = j
ki : i ⊆ s \ ⋃₀ (↑J : Set (Set α))
hx2 : j ⊆ x
⊢ j ⊆ ⋃₀ (↑J : Set (Set α))

Tactic:
hx2.trans <| subset_sUnion_of_mem hx

Premises used: HasSubset.Subset.trans, Set.subset_sUnion_of_mem
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
i : Set α
hi : i ∈ K1 s
j x : Set α
hx : x ∈ J
h3 : j ∈ K1 x
h4 : ¬i = j
ki : i ⊆ s \ ⋃₀ (↑J : Set (Set α))
hx2 : j ⊆ x
⊢ j ⊆ ⋃₀ (↑J : Set (Set α))

Tactic:
obtain kj : j ⊆ ⋃₀ J := hx2.trans <| subset_sUnion_of_mem hx

Premises used: HasSubset.Subset.trans, Set.subset_sUnion_of_mem
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
i : Set α
hi : i ∈ K1 s
j x : Set α
hx : x ∈ J
h3 : j ∈ K1 x
h4 : ¬i = j
ki : i ⊆ s \ ⋃₀ (↑J : Set (Set α))
hx2 : j ⊆ x
kj : j ⊆ ⋃₀ (↑J : Set (Set α))
⊢ Disjoint i j

Tactic:
disjoint_of_subset ki kj disjoint_sdiff_left

Premises used: Set.disjoint_of_subset, Set.disjoint_sdiff_left
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
i : Set α
hi : i ∈ K1 s
j x : Set α
hx : x ∈ J
h3 : j ∈ K1 x
h4 : ¬i = j
ki : i ⊆ s \ ⋃₀ (↑J : Set (Set α))
hx2 : j ⊆ x
kj : j ⊆ ⋃₀ (↑J : Set (Set α))
⊢ Disjoint i j

Tactic:
exact disjoint_of_subset ki kj disjoint_sdiff_left

Premises used: Set.disjoint_of_subset, Set.disjoint_sdiff_left
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
a : Set α
ha : a ∈ J
h : ∀ t' ∈ K a, t' ⊆ a
⊢ ∀ t' ∈ K1 a, t' ⊆ a

Tactic:
simp_rw [hK1_of_ne _ (ne_of_mem_of_not_mem ha hJ)]

Premises used: ne_of_mem_of_not_mem
-- Module: Mathlib.MeasureTheory.SetSemiring
Goal:
α : Type u_1
C : Set (Set α)
J_1 : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
  ∀ (h1 : (↑J : Set (Set α)) ⊆ C),
    ∃ (K : Set α → Finset (Set α)),
      (↑J : Set (Set α)).PairwiseDisjoint K ∧
        (∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C) ∧
          PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id ∧
            (∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j) ∧
              (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
h1 : s ∈ C ∧ (↑J : Set (Set α)) ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J : Set (Set α)).PairwiseDisjoint K
hK1 : ∀ i ∈ J, (↑(K i) : Set (Set α)) ⊆ C
hK2 : PairwiseDisjoint.{u_1, u_1} (α := Set α) (⋃ x ∈ J, (↑(K x) : Set (Set α))) id
hK3 : ∀ j ∈ J, ⋃₀ (↑(K j) : Set (Set α)) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ ⋃ x ∈ J, (↑(K x) : Set (Set α))
K1 : Set α → Finset (Set α) := fun (t : Set α) ↦ if t = s then hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion (C := C) (s := s) ⋯ ⋯
hK1_of_ne : ∀ (t : Set α) (ht : t ≠ s), K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : Eq.{u_1 + 1} (α := Set (Set α)) (⋃ x ∈ J, (↑(K1 x) : Set (Set α))) (⋃ x ∈ J, (↑(K x) : Set (Set α)))
a : Set α
ha : a ∈ J
h : ⋃₀ (↑(K a) : Set (Set α)) ⊆ a
⊢ ∀ t' ∈ (↑(K a) : Set (Set α)), t' ⊆ a

Tactic:
rw [← sUnion_subset_iff]

Premises used: Set.sUnion_subset_iff