Theorem: MeasureTheory.IsSetSemiring.exists_disjoint_finset_diff_eq
File: Mathlib/MeasureTheory/SetSemiring.lean

tactic: induction I using Finset.induction with
| empty =>
  simp only [coe_empty, sUnion_empty, diff_empty, exists_prop]
  refine ⟨{s}, singleton_subset_set_iff.mpr hs, ?_⟩
  simp only [coe_singleton, pairwiseDisjoint_singleton, sUnion_singleton, eq_self_iff_true,
    and_self_iff]
| @insert t I' _ h => ?_
state_before: α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I = ⋃₀ ↑J
state_after: case insert
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : ↑(insert t I') ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: rw [coe_insert] at hI
state_before: case insert
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : ↑(insert t I') ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: have ht : t ∈ C := hI (Set.mem_insert _ _)
state_before: case insert
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: obtain ⟨J, h_ss, h_dis, h_eq⟩ := h ((Set.subset_insert _ _).trans hI)
state_before: case insert
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: let Ju : ∀ u ∈ C, Finset (Set α) := fun u hu ↦ hC.diffFinset hu ht
state_before: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: have hJu_subset : ∀ (u) (hu : u ∈ C), ↑(Ju u hu) ⊆ C := by
  intro u hu x hx
  exact hC.diffFinset_subset hu ht hx
state_before: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: have hJu_disj : ∀ (u) (hu : u ∈ C), (Ju u hu : Set (Set α)).PairwiseDisjoint id := fun u hu ↦
  hC.pairwiseDisjoint_diffFinset hu ht
state_before: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: have hJu_sUnion : ∀ (u) (hu : u ∈ C), ⋃₀ (Ju u hu : Set (Set α)) = u \ t :=
  fun u hu ↦ hC.sUnion_diffFinset hu ht
state_before: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: 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
state_before: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: let J' : Finset (Set α) := Finset.biUnion (Finset.univ : Finset J) fun u ↦ Ju u (h_ss u.prop)
state_before: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: have hJ'_subset : ↑J' ⊆ C := by
  intro u
  simp only [J' ,Subtype.coe_mk, 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
state_before: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J

tactic: refine ⟨J', hJ'_subset, ?_, ?_⟩
state_before: case insert.intro.intro.intro
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
state_after: case insert.intro.intro.intro.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ (↑J').PairwiseDisjoint id

case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J'

tactic: simp only [coe_empty, sUnion_empty, diff_empty, exists_prop]
state_before: case empty
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑∅ ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑∅ = ⋃₀ ↑J
state_after: case empty
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑∅ ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s = ⋃₀ ↑J

tactic: refine ⟨{s}, singleton_subset_set_iff.mpr hs, ?_⟩
state_before: case empty
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑∅ ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s = ⋃₀ ↑J
state_after: case empty
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑∅ ⊆ C
⊢ (↑{s}).PairwiseDisjoint id ∧ s = ⋃₀ ↑{s}

tactic: simp only [coe_singleton, pairwiseDisjoint_singleton, sUnion_singleton, eq_self_iff_true,
  and_self_iff]
state_before: case empty
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑∅ ⊆ C
⊢ (↑{s}).PairwiseDisjoint id ∧ s = ⋃₀ ↑{s}
state_after: no goals

tactic: intro u hu x hx
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
⊢ ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
u : Set α
hu : u ∈ C
x : Set α
hx : x ∈ ↑(Ju u hu)
⊢ x ∈ C

tactic: exact hC.diffFinset_subset hu ht hx
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
u : Set α
hu : u ∈ C
x : Set α
hx : x ∈ ↑(Ju u hu)
⊢ x ∈ C
state_after: no goals

tactic: intro u hu v hv huv_disj
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
⊢ ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
u : Set α
hu : u ∈ C
v : Set α
hv : v ∈ C
huv_disj : Disjoint u v
⊢ Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))

tactic: rw [hJu_sUnion, hJu_sUnion]
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
u : Set α
hu : u ∈ C
v : Set α
hv : v ∈ C
huv_disj : Disjoint u v
⊢ Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
u : Set α
hu : u ∈ C
v : Set α
hv : v ∈ C
huv_disj : Disjoint u v
⊢ Disjoint (u \ t) (v \ t)

tactic: exact disjoint_of_subset Set.diff_subset Set.diff_subset huv_disj
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
u : Set α
hu : u ∈ C
v : Set α
hv : v ∈ C
huv_disj : Disjoint u v
⊢ Disjoint (u \ t) (v \ t)
state_after: no goals

tactic: intro u
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
⊢ ↑J' ⊆ C
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
u : Set α
⊢ u ∈ ↑J' → u ∈ C

tactic: simp only [J' ,Subtype.coe_mk, univ_eq_attach, coe_biUnion, mem_coe, mem_attach, iUnion_true,
  mem_iUnion, Finset.exists_coe, exists₂_imp]
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
u : Set α
⊢ u ∈ ↑J' → u ∈ C
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
u : Set α
⊢ ∀ (x : Set α) (h : x ∈ J), u ∈ Ju x ⋯ → u ∈ C

tactic: intro v hv huvt
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
u : Set α
⊢ ∀ (x : Set α) (h : x ∈ J), u ∈ Ju x ⋯ → u ∈ C
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
u v : Set α
hv : v ∈ J
huvt : u ∈ Ju v ⋯
⊢ u ∈ C

tactic: exact hJu_subset v (h_ss hv) huvt
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
u v : Set α
hv : v ∈ J
huvt : u ∈ Ju v ⋯
⊢ u ∈ C
state_after: no goals

tactic: rw [Finset.coe_biUnion]
state_before: case insert.intro.intro.intro.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ (↑J').PairwiseDisjoint id
state_after: case insert.intro.intro.intro.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ (⋃ x ∈ ↑Finset.univ, ↑(Ju ↑x ⋯)).PairwiseDisjoint id

tactic: refine PairwiseDisjoint.biUnion ?_ ?_
state_before: case insert.intro.intro.intro.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ (⋃ x ∈ ↑Finset.univ, ↑(Ju ↑x ⋯)).PairwiseDisjoint id
state_after: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ (↑Finset.univ).PairwiseDisjoint fun i' => ⨆ i ∈ ↑(Ju ↑i' ⋯), id i

case insert.intro.intro.intro.refine_1.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ∀ i ∈ ↑Finset.univ, (↑(Ju ↑i ⋯)).PairwiseDisjoint id

tactic: simp only [univ_eq_attach, mem_coe, id, iSup_eq_iUnion]
state_before: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ (↑Finset.univ).PairwiseDisjoint fun i' => ⨆ i ∈ ↑(Ju ↑i' ⋯), id i
state_after: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ (↑J.attach).PairwiseDisjoint fun i' => ⋃ i ∈ Ju ↑i' ⋯, i

tactic: simp_rw [PairwiseDisjoint, Set.Pairwise]
state_before: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ (↑J.attach).PairwiseDisjoint fun i' => ⋃ i ∈ Ju ↑i' ⋯, i
state_after: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ∀ ⦃x : { x // x ∈ J }⦄,
    x ∈ ↑J.attach → ∀ ⦃y : { x // x ∈ J }⦄, y ∈ ↑J.attach → x ≠ y → (Disjoint on fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x y

tactic: intro x _ y _ hxy
state_before: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ∀ ⦃x : { x // x ∈ J }⦄,
    x ∈ ↑J.attach → ∀ ⦃y : { x // x ∈ J }⦄, y ∈ ↑J.attach → x ≠ y → (Disjoint on fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x y
state_after: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
⊢ (Disjoint on fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x y

tactic: 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
state_before: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
⊢ (Disjoint on fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x y
state_after: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (Disjoint on fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x y

tactic: convert hJu_disj' (x : Set α) (h_ss x.prop) y (h_ss y.prop) hxy_disj
state_before: case insert.intro.intro.intro.refine_1.refine_1
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (Disjoint on fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x y
state_after: case h.e'_4
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x = ⋃₀ ↑(Ju ↑x ⋯)

case h.e'_5
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) y = ⋃₀ ↑(Ju ↑y ⋯)

tactic: by_contra h_contra
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
⊢ Disjoint ↑x ↑y
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
h_contra : ¬Disjoint ↑x ↑y
⊢ False

tactic: refine hxy ?_
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
h_contra : ¬Disjoint ↑x ↑y
⊢ False
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
h_contra : ¬Disjoint ↑x ↑y
⊢ x = y

tactic: refine Subtype.ext ?_
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
h_contra : ¬Disjoint ↑x ↑y
⊢ x = y
state_after: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
h_contra : ¬Disjoint ↑x ↑y
⊢ ↑x = ↑y

tactic: exact h_dis.elim x.prop y.prop h_contra
state_before: α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
h_contra : ¬Disjoint ↑x ↑y
⊢ ↑x = ↑y
state_after: no goals

tactic: rw [sUnion_eq_biUnion]
state_before: case h.e'_4
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x = ⋃₀ ↑(Ju ↑x ⋯)
state_after: case h.e'_4
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x = ⋃ i ∈ ↑(Ju ↑x ⋯), i

tactic: congr
state_before: case h.e'_4
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) x = ⋃ i ∈ ↑(Ju ↑x ⋯), i
state_after: no goals

tactic: rw [sUnion_eq_biUnion]
state_before: case h.e'_5
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) y = ⋃₀ ↑(Ju ↑y ⋯)
state_after: case h.e'_5
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) y = ⋃ i ∈ ↑(Ju ↑y ⋯), i

tactic: congr
state_before: case h.e'_5
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝² : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
x : { x // x ∈ J }
a✝¹ : x ∈ ↑J.attach
y : { x // x ∈ J }
a✝ : y ∈ ↑J.attach
hxy : x ≠ y
hxy_disj : Disjoint ↑x ↑y
⊢ (fun i' => ⋃ i ∈ Ju ↑i' ⋯, i) y = ⋃ i ∈ ↑(Ju ↑y ⋯), i
state_after: no goals

tactic: exact fun u _ ↦ hJu_disj _ _
state_before: case insert.intro.intro.intro.refine_1.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ∀ i ∈ ↑Finset.univ, (↑(Ju ↑i ⋯)).PairwiseDisjoint id
state_after: no goals

tactic: rw [coe_insert, sUnion_insert, Set.union_comm, ← Set.diff_diff, h_eq]
state_before: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J'
state_after: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ⋃₀ ↑J \ t = ⋃₀ ↑J'

tactic: simp_rw [J', sUnion_eq_biUnion, Set.iUnion_diff]
state_before: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ⋃₀ ↑J \ t = ⋃₀ ↑J'
state_after: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ⋃ i ∈ ↑J, i \ t = ⋃ i ∈ ↑(Finset.univ.biUnion fun u => Ju ↑u ⋯), i

tactic: simp only [Subtype.coe_mk, mem_coe, Finset.mem_biUnion, Finset.mem_univ, exists_true_left,
  Finset.exists_coe, iUnion_exists, true_and]
state_before: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ⋃ i ∈ ↑J, i \ t = ⋃ i ∈ ↑(Finset.univ.biUnion fun u => Ju ↑u ⋯), i
state_after: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ⋃ i ∈ J, i \ t = ⋃ i, ⋃ i_1, ⋃ (i_2 : i_1 ∈ J), ⋃ (_ : i ∈ Ju i_1 ⋯), i

tactic: rw [iUnion_comm]
state_before: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ⋃ i ∈ J, i \ t = ⋃ i, ⋃ i_1, ⋃ (i_2 : i_1 ∈ J), ⋃ (_ : i ∈ Ju i_1 ⋯), i
state_after: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ⋃ i ∈ J, i \ t = ⋃ i', ⋃ i, ⋃ (i_1 : i' ∈ J), ⋃ (_ : i ∈ Ju i' ⋯), i

tactic: refine iUnion_congr fun i ↦ ?_
state_before: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
⊢ ⋃ i ∈ J, i \ t = ⋃ i', ⋃ i, ⋃ (i_1 : i' ∈ J), ⋃ (_ : i ∈ Ju i' ⋯), i
state_after: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
⊢ ⋃ (_ : i ∈ J), i \ t = ⋃ i_1, ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1

tactic: by_cases hi : i ∈ J
state_before: case insert.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
⊢ ⋃ (_ : i ∈ J), i \ t = ⋃ i_1, ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1
state_after: case pos
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
hi : i ∈ J
⊢ ⋃ (_ : i ∈ J), i \ t = ⋃ i_1, ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1

case neg
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
hi : i ∉ J
⊢ ⋃ (_ : i ∈ J), i \ t = ⋃ i_1, ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1

tactic: simp only [hi, iUnion_true, exists_prop]
state_before: case pos
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
hi : i ∈ J
⊢ ⋃ (_ : i ∈ J), i \ t = ⋃ i_1, ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1
state_after: case pos
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
hi : i ∈ J
⊢ i \ t = ⋃ i_1 ∈ Ju i ⋯, i_1

tactic: rw [← hJu_sUnion i (h_ss hi), sUnion_eq_biUnion]
state_before: case pos
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
hi : i ∈ J
⊢ i \ t = ⋃ i_1 ∈ Ju i ⋯, i_1
state_after: case pos
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
hi : i ∈ J
⊢ ⋃ i_1 ∈ ↑(Ju i ⋯), i_1 = ⋃ i_1 ∈ Ju i ⋯, i_1

tactic: simp only [mem_coe]
state_before: case pos
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
hi : i ∈ J
⊢ ⋃ i_1 ∈ ↑(Ju i ⋯), i_1 = ⋃ i_1 ∈ Ju i ⋯, i_1
state_after: no goals

tactic: simp only [hi, iUnion_of_empty, iUnion_empty]
state_before: case neg
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => hC.diffFinset hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), (↑(Ju u hu)).PairwiseDisjoint id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.univ.biUnion fun u => Ju ↑u ⋯
hJ'_subset : ↑J' ⊆ C
i : Set α
hi : i ∉ J
⊢ ⋃ (_ : i ∈ J), i \ t = ⋃ i_1, ⋃ (i_2 : i ∈ J), ⋃ (_ : i_1 ∈ Ju i ⋯), i_1
state_after: no goals