-- 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
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))
insert insert₁ :
  ∀ (t : Set α) (I' : Finset (Set α)),
    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.insert t I') : Set (Set α)) ⊆ C),
        ∃ (J : Finset (Set α)),
          (↑J : Set (Set α)) ⊆ C ∧
            (↑J : Set (Set α)).PairwiseDisjoint id ∧
              s \ ⋃₀ (↑(Insert.insert t I') : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
insert₂ :
  ∀ (t : Set α) (I' : Finset (Set α)),
    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.insert t (↑I' : Set (Set α)) ⊆ C),
        ∃ (J : Finset (Set α)),
          (↑J : Set (Set α)) ⊆ C ∧
            (↑J : Set (Set α)).PairwiseDisjoint id ∧
              s \ ⋃₀ (↑(Insert.insert t I') : Set (Set α)) = ⋃₀ (↑J : Set (Set α))
⊢ ∃ (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 insert₃ : ∀ (t : Set α) (I' : Finset (Set α)),
  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.insert t (↑I' : Set (Set α)) ⊆ C) (ht : t ∈ C),
      ∃ (J : Finset (Set α)),
        (↑J : Set (Set α)) ⊆ C ∧
          (↑J : Set (Set α)).PairwiseDisjoint id ∧ s \ ⋃₀ (↑(Insert.insert t I') : 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