/-- A finite family of ideals is pairwise coprime (that is, any two of them generate the whole ring)
iff when taking all the possible intersections of all but one of these ideals, the resulting family
of ideals still generate the whole ring.

For example with three ideals : `I ⊔ J = I ⊔ K = J ⊔ K = ⊤ ↔ (I ⊓ J) ⊔ (I ⊓ K) ⊔ (J ⊓ K) = ⊤`.

When ideals are all of the form `I i = R ∙ s i`, this is equivalent to the
`exists_sum_eq_one_iff_pairwise_coprime` lemma.-/
theorem iSup_iInf_eq_top_iff_pairwise {t : Finset ι} (h : t.Nonempty) (I : ι → Ideal R) :
    (⨆ i ∈ t, ⨅ (j) (_ : j ∈ t) (_ : j ≠ i), I j) = ⊤ ↔
      (t : Set ι).Pairwise fun i j => I i ⊔ I j = ⊤ := by
  haveI : DecidableEq ι := Classical.decEq ι
  rw [eq_top_iff_one, Submodule.mem_iSup_finset_iff_exists_sum]
  refine' h.cons_induction _ _ <;> clear t h
  · simp only [Finset.sum_singleton, Finset.coe_singleton, Set.pairwise_singleton, iff_true_iff]
    refine' fun a => ⟨fun i => if h : i = a then ⟨1, _⟩ else 0, _⟩
    · simp [h]
    · simp only [dif_pos, dif_ctx_congr, Submodule.coe_mk, eq_self_iff_true]
  intro a t hat h ih
  rw [Finset.coe_cons,
    Set.pairwise_insert_of_symmetric fun i j (h : I i ⊔ I j = ⊤) => sup_comm.trans h]
  constructor
  · rintro ⟨μ, hμ⟩
    rw [Finset.sum_cons] at hμ
    -- Porting note: `refine` yields goals in a different order than in lean3.
    refine ⟨ih.mp ⟨Pi.single h.choose ⟨μ a, ?a1⟩ + fun i => ⟨μ i, ?a2⟩, ?a3⟩, fun b hb ab => ?a4⟩
    case a1 =>
      have := Submodule.coe_mem (μ a)
      rw [mem_iInf] at this ⊢
      --for some reason `simp only [mem_iInf]` times out
      intro i
      specialize this i
      rw [mem_iInf, mem_iInf] at this ⊢
      intro hi _
      apply this (Finset.subset_cons _ hi)
      rintro rfl
      exact hat hi
    

Proof state:
case a3
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : Finset.Nonempty t
ih : (∃ μ, (Finset.sum t fun i => ↑(μ i)) = 1) ↔ Set.Pairwise ↑t fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : (↑(μ a) + Finset.sum t fun x => ↑(μ x)) = 1
⊢ (Finset.sum t fun i =>
      ↑((Pi.single (Exists.choose h)
              { val := ↑(μ a), property := (_ : ↑(μ a) ∈ ⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose h), I j) } +
            fun i => ?m.8523 i)
          i)) =
    1
case a4
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : Finset.Nonempty t
ih : (∃ μ, (Finset.sum t fun i => ↑(μ i)) = 1) ↔ Set.Pairwise ↑t fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : (↑(μ a) + Finset.sum t fun x => ↑(μ x)) = 1
b : ι
hb : b ∈ ↑t
ab : a ≠ b
⊢ I a ⊔ I b = ⊤
case a2
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : Finset.Nonempty t
ih : (∃ μ, (Finset.sum t fun i => ↑(μ i)) = 1) ↔ Set.Pairwise ↑t fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : (↑(μ a) + Finset.sum t fun x => ↑(μ x)) = 1
i : ι
⊢ ↑(μ i) ∈ ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j

Next tactic:
case a2 =>
      have := Submodule.coe_mem (μ i)
      simp only [mem_iInf] at this ⊢
      intro j hj ij
      exact this _ (Finset.subset_cons _ hj) ij