-- Mathlib/RingTheory/Coprime/Ideal.lean
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 = ⊤

Proof state:
case refine_2.mpr.intro.intro
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
hs : (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
Hb : ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤
μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)
hμ : ∑ i ∈ t, ↑(μ i) = 1
⊢ ∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1

Tactic:
have := sup_iInf_eq_top fun b hb => Hb b hb (ne_of_mem_of_not_mem hb hat).symm

Result:
case refine_2.mpr.intro.intro
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
hs : (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
Hb : ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤
μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)
hμ : ∑ i ∈ t, ↑(μ i) = 1
this : I a ⊔ ⨅ i ∈ t, I i = ⊤
⊢ ∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1

Proof state:
case h.e'_2.a
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ✝ : ↑(μ a) + ∑ x ∈ t, ↑(μ x) = 1
hμ : ∑ x ∈ t, (Pi.single (Exists.choose h) (↑(μ a)) x + ↑(μ x)) = 1
i : ι
a✝ : i ∈ t
⊢ ↑(Pi.single (Exists.choose h) ⟨↑(μ a), ⋯⟩ i) + ↑⟨↑(μ i), ⋯⟩ = Pi.single (Exists.choose h) (↑(μ a)) i + ↑(μ i)

Tactic:
by_cases hi : i = h.choose

Result:
case pos
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ✝ : ↑(μ a) + ∑ x ∈ t, ↑(μ x) = 1
hμ : ∑ x ∈ t, (Pi.single (Exists.choose h) (↑(μ a)) x + ↑(μ x)) = 1
i : ι
a✝ : i ∈ t
hi : i = Exists.choose h
⊢ ↑(Pi.single (Exists.choose h) ⟨↑(μ a), ⋯⟩ i) + ↑⟨↑(μ i), ⋯⟩ = Pi.single (Exists.choose h) (↑(μ a)) i + ↑(μ i)

case neg
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ✝ : ↑(μ a) + ∑ x ∈ t, ↑(μ x) = 1
hμ : ∑ x ∈ t, (Pi.single (Exists.choose h) (↑(μ a)) x + ↑(μ x)) = 1
i : ι
a✝ : i ∈ t
hi : ¬i = Exists.choose h
⊢ ↑(Pi.single (Exists.choose h) ⟨↑(μ a), ⋯⟩ i) + ↑⟨↑(μ i), ⋯⟩ = Pi.single (Exists.choose h) (↑(μ a)) i + ↑(μ i)

Proof state:
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ↑(μ a) + ∑ x ∈ t, ↑(μ x) = 1
i : ι
this : ↑(μ i) ∈ ⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j
⊢ ↑(μ i) ∈ ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j

Tactic:
simp only [mem_iInf] at this ⊢

Result:
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ↑(μ a) + ∑ x ∈ t, ↑(μ x) = 1
i : ι
this : ∀ i_1 ∈ Finset.cons a t hat, i_1 ≠ i → ↑(μ i) ∈ I i_1
⊢ ∀ i_1 ∈ t, i_1 ≠ i → ↑(μ i) ∈ I i_1

Proof state:
case refine_2
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, ↑(μ x) + ↑(μ a) = 1
b : ι
hb : b ∈ ↑t
ab : a ≠ b
this : ↑(μ a) ∈ ⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ a), I j
⊢ ↑(μ a) ∈ I b

Tactic:
simp only [mem_iInf] at this

Result:
case refine_2
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, ↑(μ x) + ↑(μ a) = 1
b : ι
hb : b ∈ ↑t
ab : a ≠ b
this : ∀ i ∈ Finset.cons a t hat, i ≠ a → ↑(μ a) ∈ I i
⊢ ↑(μ a) ∈ I b

Proof state:
case refine_1
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, ↑(μ x) + ↑(μ a) = 1
b : ι
hb : b ∈ ↑t
ab : a ≠ b
x : ι
hx : x ∈ t
this : ∀ i ∈ Finset.cons a t hat, i ≠ x → ↑(μ x) ∈ I i
⊢ a ≠ x

Tactic:
rintro rfl

Result:
case refine_1
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, ↑(μ x) + ↑(μ a) = 1
b : ι
hb : b ∈ ↑t
ab : a ≠ b
hx : a ∈ t
this : ∀ i ∈ Finset.cons a t hat, i ≠ a → ↑(μ a) ∈ I i
⊢ False

Proof state:
case refine_2.mp
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) →
    ((↑t).Pairwise fun i j => I i ⊔ I j = ⊤) ∧ ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤

Tactic:
rintro ⟨μ, hμ⟩

Result:
case refine_2.mp.intro
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1
⊢ ((↑t).Pairwise fun i j => I i ⊔ I j = ⊤) ∧ ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤

Proof state:
case neg
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ✝ : ↑(μ a) + ∑ x ∈ t, ↑(μ x) = 1
hμ : ∑ x ∈ t, (Pi.single (Exists.choose h) (↑(μ a)) x + ↑(μ x)) = 1
i : ι
a✝ : i ∈ t
hi : ¬i = Exists.choose h
⊢ ↑(Pi.single (Exists.choose h) ⟨↑(μ a), ⋯⟩ i) + ↑⟨↑(μ i), ⋯⟩ = Pi.single (Exists.choose h) (↑(μ a)) i + ↑(μ i)

Tactic:
rw [Pi.single_eq_of_ne hi, Pi.single_eq_of_ne hi, Submodule.coe_zero]

Result:
no goals

Proof state:
case refine_2
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) ↔ (↑(Finset.cons a t hat)).Pairwise fun i j => I i ⊔ I j = ⊤

Tactic:
rw [Finset.coe_cons,
  Set.pairwise_insert_of_symmetric fun i j (h : I i ⊔ I j = ⊤) ↦ (sup_comm _ _).trans h]

Result:
case refine_2
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 : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) ↔
    ((↑t).Pairwise fun i j => I i ⊔ I j = ⊤) ∧ ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤