awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
h_1 : t ∉ I'
h :
  ∀ (hI : (↑I' : Set (Set α)) ⊆ C),
    ∃ (J : Finset (Set α)),
      (↑J : Set (Set α)) ⊆ C ∧
        (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
hI : insert t (↑I' : Set (Set α)) ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_dis : (↑J : Set (Set α)).PairwiseDisjoint id
h_eq : s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Ju : (u : Set α) → u ∈ C → Finset (Set α) :=
  fun (u : Set α) (hu : u ∈ C) ↦ hC.disjointOfDiff (C := C) (s := u) (t := t) hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ (↑(Ju u hu) : Set (Set α)) = u \ t
hJu_disj' :
  ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C) (_h_dis : Disjoint u v),
    Disjoint (⋃₀ (↑(Ju u hu) : Set (Set α))) (⋃₀ (↑(Ju v hv) : Set (Set α)))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯
hJ'_subset : (↑J' : Set (Set α)) ⊆ C
⊢ ⋃₀ (↑J : Set (Set α)) \ t = ⋃₀ (↑J' : Set (Set α))
have insert : Eq.{u_1 + 1} (α := Set α) (⋃ i ∈ (↑J : Set (Set α)), i \ t)
  (⋃ i ∈ (↑(Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) sorry) : Set (Set α)), i)
Tactic: simp_rw [J', sUnion_eq_biUnion, Set.iUnion_diff]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
h_1 : t ∉ I'
h :
  ∀ (hI : (↑I' : Set (Set α)) ⊆ C),
    ∃ (J : Finset (Set α)),
      (↑J : Set (Set α)) ⊆ C ∧
        (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
hI : insert t (↑I' : Set (Set α)) ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_dis : (↑J : Set (Set α)).PairwiseDisjoint id
h_eq : s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Ju : (u : Set α) → u ∈ C → Finset (Set α) :=
  fun (u : Set α) (hu : u ∈ C) ↦ hC.disjointOfDiff (C := C) (s := u) (t := t) hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ (↑(Ju u hu) : Set (Set α)) = u \ t
hJu_disj' :
  ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C) (_h_dis : Disjoint u v),
    Disjoint (⋃₀ (↑(Ju u hu) : Set (Set α))) (⋃₀ (↑(Ju v hv) : Set (Set α)))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯
hJ'_subset : (↑J' : Set (Set α)) ⊆ C
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ i ∈ (↑J : Set (Set α)), i \ t)
    (⋃ i ∈ (↑(Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯) : Set (Set α)), i)
have insert : Eq.{u_1 + 1} (α := Set α) (⋃ i ∈ J, i \ t) (⋃ (i : Set α), ⋃ (i_1 : Set α), ⋃ (i_2 : i_1 ∈ J), ⋃ (_ : i ∈ Ju i_1 sorry), i)
Tactic: simp only [mem_coe, Finset.mem_biUnion, Finset.mem_univ,
      Finset.exists_coe, iUnion_exists, true_and]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
h_1 : t ∉ I'
h :
  ∀ (hI : (↑I' : Set (Set α)) ⊆ C),
    ∃ (J : Finset (Set α)),
      (↑J : Set (Set α)) ⊆ C ∧
        (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
hI : insert t (↑I' : Set (Set α)) ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_dis : (↑J : Set (Set α)).PairwiseDisjoint id
h_eq : s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Ju : (u : Set α) → u ∈ C → Finset (Set α) :=
  fun (u : Set α) (hu : u ∈ C) ↦ hC.disjointOfDiff (C := C) (s := u) (t := t) hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ (↑(Ju u hu) : Set (Set α)) = u \ t
hJu_disj' :
  ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C) (_h_dis : Disjoint u v),
    Disjoint (⋃₀ (↑(Ju u hu) : Set (Set α))) (⋃₀ (↑(Ju v hv) : Set (Set α)))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯
hJ'_subset : (↑J' : Set (Set α)) ⊆ C
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ i ∈ J, i \ t)
    (⋃ (i : Set α), ⋃ (i_1 : Set α), ⋃ (i_2 : i_1 ∈ J), ⋃ (_ : i ∈ Ju i_1 ⋯), i)
have insert : Eq.{u_1 + 1} (α := Set α) (⋃ i ∈ J, i \ t) (⋃ (i' : Set α), ⋃ (i : Set α), ⋃ (i_1 : i' ∈ J), ⋃ (_ : i ∈ Ju i' sorry), i)
Tactic: rw [iUnion_comm]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
h_1 : t ∉ I'
h :
  ∀ (hI : (↑I' : Set (Set α)) ⊆ C),
    ∃ (J : Finset (Set α)),
      (↑J : Set (Set α)) ⊆ C ∧
        (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
hI : insert t (↑I' : Set (Set α)) ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_dis : (↑J : Set (Set α)).PairwiseDisjoint id
h_eq : s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Ju : (u : Set α) → u ∈ C → Finset (Set α) :=
  fun (u : Set α) (hu : u ∈ C) ↦ hC.disjointOfDiff (C := C) (s := u) (t := t) hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ (↑(Ju u hu) : Set (Set α)) = u \ t
hJu_disj' :
  ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C) (_h_dis : Disjoint u v),
    Disjoint (⋃₀ (↑(Ju u hu) : Set (Set α))) (⋃₀ (↑(Ju v hv) : Set (Set α)))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯
hJ'_subset : (↑J' : Set (Set α)) ⊆ C
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ i ∈ J, i \ t) (⋃ (i' : Set α), ⋃ (i : Set α), ⋃ (i_1 : i' ∈ J), ⋃ (_ : i ∈ Ju i' ⋯), i)
have insert : ∀ (i : Set α),
  Eq.{u_1 + 1} (α := Set α) (⋃ (_ : i ∈ J), i \ t) (⋃ (i_1 : Set α), ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i sorry), i_1)
Tactic: refine iUnion_congr fun i ↦ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
h_1 : t ∉ I'
h :
  ∀ (hI : (↑I' : Set (Set α)) ⊆ C),
    ∃ (J : Finset (Set α)),
      (↑J : Set (Set α)) ⊆ C ∧
        (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
hI : insert t (↑I' : Set (Set α)) ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_dis : (↑J : Set (Set α)).PairwiseDisjoint id
h_eq : s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Ju : (u : Set α) → u ∈ C → Finset (Set α) :=
  fun (u : Set α) (hu : u ∈ C) ↦ hC.disjointOfDiff (C := C) (s := u) (t := t) hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ (↑(Ju u hu) : Set (Set α)) = u \ t
hJu_disj' :
  ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C) (_h_dis : Disjoint u v),
    Disjoint (⋃₀ (↑(Ju u hu) : Set (Set α))) (⋃₀ (↑(Ju v hv) : Set (Set α)))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯
hJ'_subset : (↑J' : Set (Set α)) ⊆ C
i : Set α
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ (_ : i ∈ J), i \ t) (⋃ (i_1 : Set α), ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1)
have pos : ∀ (hi : i ∈ J),
  Eq.{u_1 + 1} (α := Set α) (⋃ (_ : i ∈ J), i \ t) (⋃ (i_1 : Set α), ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i sorry), i_1)
Tactic: by_cases hi : i ∈ J
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
h_1 : t ∉ I'
h :
  ∀ (hI : (↑I' : Set (Set α)) ⊆ C),
    ∃ (J : Finset (Set α)),
      (↑J : Set (Set α)) ⊆ C ∧
        (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
hI : insert t (↑I' : Set (Set α)) ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_dis : (↑J : Set (Set α)).PairwiseDisjoint id
h_eq : s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Ju : (u : Set α) → u ∈ C → Finset (Set α) :=
  fun (u : Set α) (hu : u ∈ C) ↦ hC.disjointOfDiff (C := C) (s := u) (t := t) hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ (↑(Ju u hu) : Set (Set α)) = u \ t
hJu_disj' :
  ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C) (_h_dis : Disjoint u v),
    Disjoint (⋃₀ (↑(Ju u hu) : Set (Set α))) (⋃₀ (↑(Ju v hv) : Set (Set α)))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯
hJ'_subset : (↑J' : Set (Set α)) ⊆ C
i : Set α
pos :
  ∀ (hi : i ∈ J),
    Eq.{u_1 + 1} (α := Set α) (⋃ (_ : i ∈ J), i \ t) (⋃ (i_1 : Set α), ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1)
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ (_ : i ∈ J), i \ t) (⋃ (i_1 : Set α), ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1)
have neg : ∀ (hi : i ∉ J),
  Eq.{u_1 + 1} (α := Set α) (⋃ (_ : i ∈ J), i \ t) (⋃ (i_1 : Set α), ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i sorry), i_1)
Tactic: by_cases hi : i ∈ J
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
h_1 : t ∉ I'
h :
  ∀ (hI : (↑I' : Set (Set α)) ⊆ C),
    ∃ (J : Finset (Set α)),
      (↑J : Set (Set α)) ⊆ C ∧
        (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
hI : insert t (↑I' : Set (Set α)) ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_dis : (↑J : Set (Set α)).PairwiseDisjoint id
h_eq : s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Ju : (u : Set α) → u ∈ C → Finset (Set α) :=
  fun (u : Set α) (hu : u ∈ C) ↦ hC.disjointOfDiff (C := C) (s := u) (t := t) hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ (↑(Ju u hu) : Set (Set α)) = u \ t
hJu_disj' :
  ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C) (_h_dis : Disjoint u v),
    Disjoint (⋃₀ (↑(Ju u hu) : Set (Set α))) (⋃₀ (↑(Ju v hv) : Set (Set α)))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯
hJ'_subset : (↑J' : Set (Set α)) ⊆ C
i : Set α
hi : i ∈ J
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ (_ : i ∈ J), i \ t) (⋃ (i_1 : Set α), ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1)
have pos : i \ t = ⋃ i_1 ∈ Ju i sorry, i_1
Tactic: simp only [hi, iUnion_true]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
h_1 : t ∉ I'
h :
  ∀ (hI : (↑I' : Set (Set α)) ⊆ C),
    ∃ (J : Finset (Set α)),
      (↑J : Set (Set α)) ⊆ C ∧
        (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
hI : insert t (↑I' : Set (Set α)) ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_dis : (↑J : Set (Set α)).PairwiseDisjoint id
h_eq : s \ ⋃₀ (↑I' : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Ju : (u : Set α) → u ∈ C → Finset (Set α) :=
  fun (u : Set α) (hu : u ∈ C) ↦ hC.disjointOfDiff (C := C) (s := u) (t := t) hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu) : Set (Set α)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ (↑(Ju u hu) : Set (Set α)) = u \ t
hJu_disj' :
  ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C) (_h_dis : Disjoint u v),
    Disjoint (⋃₀ (↑(Ju u hu) : Set (Set α))) (⋃₀ (↑(Ju v hv) : Set (Set α)))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun (u : ↥J) ↦ Ju (↑u : Set α) ⋯
hJ'_subset : (↑J' : Set (Set α)) ⊆ C
i : Set α
hi : i ∈ J
⊢ i \ t = ⋃ i_1 ∈ Ju i ⋯, i_1
have pos : Eq.{u_1 + 1} (α := Set α) (⋃ i_1 ∈ (↑(Ju i sorry) : Set (Set α)), i_1) (⋃ i_1 ∈ Ju i sorry, i_1)
Tactic: rw [← hJu_sUnion i (h_ss hi), sUnion_eq_biUnion]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ ∃ (J : Finset (Set α)),
    (↑J : Set (Set α)) ⊆ C ∧ (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
have empty : ∀ (hI : (↑∅ : Set (Set α)) ⊆ C),
  ∃ (J : Finset (Set α)),
    (↑J : Set (Set α)) ⊆ C ∧ (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑∅ : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
Tactic: classical
  induction I using Finset.induction with
  | empty =>
    simp only [coe_empty, sUnion_empty, diff_empty]
    refine ⟨{s}, singleton_subset_set_iff.mpr hs, ?_⟩
    simp only [coe_singleton, pairwiseDisjoint_singleton, sUnion_singleton,
      and_self_iff]
  | insert t I' _ h => ?_
  rw [coe_insert] at hI
  have ht : t ∈ C := hI (Set.mem_insert _ _)
  obtain ⟨J, h_ss, h_dis, h_eq⟩ := h ((Set.subset_insert _ _).trans hI)
  let Ju : ∀ u ∈ C, Finset (Set α) := fun u hu ↦ hC.disjointOfDiff hu ht
  have hJu_subset : ∀ (u) (hu : u ∈ C), ↑(Ju u hu) ⊆ C := by
    intro u hu x hx
    exact hC.subset_disjointOfDiff hu ht hx
  have hJu_disj : ∀ (u) (hu : u ∈ C), (Ju u hu : Set (Set α)).PairwiseDisjoint id := fun u hu ↦
    hC.pairwiseDisjoint_disjointOfDiff hu ht
  have hJu_sUnion : ∀ (u) (hu : u ∈ C), ⋃₀ (Ju u hu : Set (Set α)) = u \ t :=
    fun u hu ↦ hC.sUnion_disjointOfDiff hu ht
  have hJu_disj' : ∀ (u) (hu : u ∈ C) (v) (hv : v ∈ C) (_h_dis : Disjoint u v),
      Disjoint (⋃₀ (Ju u hu : Set (Set α))) (⋃₀ ↑(Ju v hv)) := by
    intro u hu v hv huv_disj
    rw [hJu_sUnion, hJu_sUnion]
    exact disjoint_of_subset Set.diff_subset Set.diff_subset huv_disj
  let J' : Finset (Set α) := Finset.biUnion (Finset.univ : Finset J) fun u ↦ Ju u (h_ss u.prop)
  have hJ'_subset : ↑J' ⊆ C := by
    intro u
    simp only [J', univ_eq_attach, coe_biUnion, mem_coe, mem_attach, iUnion_true,
      mem_iUnion, Finset.exists_coe, exists₂_imp]
    intro v hv huvt
    exact hJu_subset v (h_ss hv) huvt
  refine ⟨J', hJ'_subset, ?_, ?_⟩
  · rw [Finset.coe_biUnion]
    refine PairwiseDisjoint.biUnion ?_ ?_
    · simp only [univ_eq_attach, mem_coe, id, iSup_eq_iUnion]
      simp_rw [PairwiseDisjoint, Set.Pairwise]
      intro x _ y _ hxy
      have hxy_disj : Disjoint (x : Set α) y := by
        by_contra h_contra
        refine hxy ?_
        refine Subtype.ext ?_
        exact h_dis.elim x.prop y.prop h_contra
      convert! hJu_disj' (x : Set α) (h_ss x.prop) y (h_ss y.prop) hxy_disj
      · rw [sUnion_eq_biUnion]
        congr
      · rw [sUnion_eq_biUnion]
        congr
    · exact fun u _ ↦ hJu_disj _ _
  · rw [coe_insert, sUnion_insert, Set.union_comm, ← Set.diff_diff, h_eq]
    simp_rw [J', sUnion_eq_biUnion, Set.iUnion_diff]
    simp only [mem_coe, Finset.mem_biUnion, Finset.mem_univ,
      Finset.exists_coe, iUnion_exists, true_and]
    rw [iUnion_comm]
    refine iUnion_congr fun i ↦ ?_
    by_cases hi : i ∈ J
    · simp only [hi, iUnion_true]
      rw [← hJu_sUnion i (h_ss hi), sUnion_eq_biUnion]
      simp only [mem_coe]
    · simp only [hi, iUnion_of_empty, iUnion_empty]
Cleared hypotheses: hI
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α)) ⊆ C
have simp : (↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
        (↑J : Set (Set α)) ⊆ C ∧
          (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
    Set (Set α)) ⊆
  insert ∅ C
Tactic: simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α)) ⊆ C
have simp : (↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
        (↑J : Set (Set α)) ⊆ C ∧
          (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
    Set (Set α)) ⊆
  insert ∅ C
Tactic: classical
  simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
  exact (hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.1.trans (Set.subset_insert _ _)
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α)).PairwiseDisjoint id
have simp : ((↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
            (↑J : Set (Set α)) ⊆ C ∧
              (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
        Set (Set α)) \
      {∅}).PairwiseDisjoint
  id
Tactic: simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α)).PairwiseDisjoint id
have simp : ((↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
            (↑J : Set (Set α)) ⊆ C ∧
              (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
        Set (Set α)) \
      {∅}).PairwiseDisjoint
  id
Tactic: classical
  simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton]
  exact Set.PairwiseDisjoint.subset
    (hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.2.1 diff_subset
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))
have rw : ⋃₀
    (↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
          (↑J : Set (Set α)) ⊆ C ∧
            (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
      Set (Set α)) =
  ⋃₀ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))
Tactic: rw [(hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.2.2]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ ⋃₀
      (↑(⋯.choose (p := fun (J : Finset (Set α)) ↦
            (↑J : Set (Set α)) ⊆ C ∧
              (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
        Set (Set α)) =
    ⋃₀ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))
have simp : ⋃₀
    (↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
          (↑J : Set (Set α)) ⊆ C ∧
            (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
      Set (Set α)) =
  ⋃₀
    ((↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
            (↑J : Set (Set α)) ⊆ C ∧
              (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
        Set (Set α)) \
      {∅})
Tactic: simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ ⋃₀
      (↑(⋯.choose (p := fun (J : Finset (Set α)) ↦
            (↑J : Set (Set α)) ⊆ C ∧
              (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
        Set (Set α)) =
    ⋃₀
      ((↑(⋯.choose (p := fun (J : Finset (Set α)) ↦
              (↑J : Set (Set α)) ⊆ C ∧
                (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
          Set (Set α)) \
        {∅})
have rw : ⋃₀
    (↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
          (↑J : Set (Set α)) ⊆ C ∧
            (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
      Set (Set α)) =
  ⋃₀
    (↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
          (↑J : Set (Set α)) ⊆ C ∧
            (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
      Set (Set α))
Tactic: rw [sUnion_diff_singleton_empty]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))
have rw : ⋃₀
    (↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
          (↑J : Set (Set α)) ⊆ C ∧
            (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
      Set (Set α)) =
  ⋃₀ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))
Tactic: classical
  rw [(hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.2.2]
  simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton]
  rw [sUnion_diff_singleton_empty]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
rw rw₁ rw₂ :
  ⋃₀
      (↑(⋯.choose (p := fun (J : Finset (Set α)) ↦
            (↑J : Set (Set α)) ⊆ C ∧
              (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
        Set (Set α)) =
    ⋃₀ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))
⊢ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))
have rw₃ : ⋃₀
    (↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
          (↑J : Set (Set α)) ⊆ C ∧
            (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
      Set (Set α)) =
  ⋃₀
    ((↑(sorry.choose (p := fun (J : Finset (Set α)) ↦
            (↑J : Set (Set α)) ⊆ C ∧
              (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑I : Set (Set α)) = ⋃₀ (↑J : Set (Set α)))) :
        Set (Set α)) \
      {∅})
Tactic: classical
  rw [(hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.2.2]
  simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton]
  rw [sUnion_diff_singleton_empty]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
⊢ ⋃₀ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α)) ⊆ s
have rw : s \ ⋃₀ (↑I : Set (Set α)) ⊆ s
Tactic: rw [← hC.diff_sUnion_eq_sUnion_disjointOfDiffUnion]
-- Lean 4 proof step from Mathlib.MeasureTheory.SetSemiring (goal)
Goal:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : (↑I : Set (Set α)) ⊆ C
t : Set α
ht : t ∈ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))
⊢ t ⊆ s \ ⋃₀ (↑I : Set (Set α))
have revert : ∀ (t : Set α) (ht : t ∈ (↑(hC.disjointOfDiffUnion (C := C) (s := s) hs hI) : Set (Set α))),
  t ⊆ s \ ⋃₀ (↑I : Set (Set α))
Tactic: revert t ht
Cleared hypotheses: t, ht