theorem exists_sum_eq_one_iff_pairwise_coprime [DecidableEq I] (h : t.Nonempty) :
    (∃ μ : I → R, (∑ i in t, μ i * ∏ j in t \ {i}, s j) = 1) ↔
      Pairwise (IsCoprime on fun i : t ↦ s i) := by
  refine' h.cons_induction _ _
  · simp only [sum_singleton, Finset.sdiff_self, prod_empty, mul_one, exists_apply_eq,
               Pairwise, Ne.def, true_iff_iff]
    rintro a ⟨i, hi⟩ ⟨j, hj⟩ h
    rw [Finset.mem_singleton] at hi hj
    simp [hi, hj] at h
  intro a t hat h ih
  rw [pairwise_cons']
  have mem : ∀ x ∈ t, a ∈ insert a t \ {x} := fun x hx ↦ by
    rw [mem_sdiff, mem_singleton]
    refine ⟨mem_insert_self _ _, fun ha ↦ hat (ha ▸ hx)⟩
  constructor
  · rintro ⟨μ, hμ⟩
    rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat] at hμ
    refine' ⟨ih.mp ⟨Pi.single h.choose (μ a * s h.choose) + μ * fun _ ↦ s a, ?_⟩, fun b hb ↦ ?_⟩
    · rw [prod_eq_mul_prod_diff_singleton h.choose_spec, ← mul_assoc, ←
        @if_pos _ _ h.choose_spec R (_ * _) 0, ← sum_pi_single', ← sum_add_distrib] at hμ
      rw [← hμ, sum_congr rfl]
      intro x hx
      dsimp -- porting note: terms were showing as sort of `HAdd.hadd` instead of `+`
      -- this whole proof pretty much breaks and has to be rewritten from scratch
      rw [add_mul]
      congr 1
      · by_cases hx : x = h.choose
        · rw [hx, Pi.single_eq_same, Pi.single_eq_same]
        · rw [Pi.single_eq_of_ne hx, Pi.single_eq_of_ne hx, zero_mul]
      · rw [mul_assoc]
        congr
        rw [prod_eq_prod_diff_singleton_mul (mem x hx) _, mul_comm]
        congr 2
        rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
    · have : IsCoprime (s b) (s a) :=
        ⟨μ a * ∏ i in t \ {b}, s i, ∑ i in t, μ i * ∏ j in t \ {i}, s j, ?_⟩
      · exact ⟨this.symm, this⟩
      rw [mul_assoc, ← prod_eq_prod_diff_singleton_mul hb, sum_mul, ← hμ, sum_congr rfl]
      intro x hx
      rw [mul_assoc]
      congr
      rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
      congr 2
      rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
  · rintro ⟨hs, Hb⟩
    obtain ⟨μ, hμ⟩ := ih.mpr hs
    obtain ⟨u, v, huv⟩ := IsCoprime.prod_left fun b hb ↦ (Hb b hb).right
    use fun i ↦ if i = a then u else v * μ i
    have hμ' : (∑ i in t, v * ((μ i * ∏ j in t \ {i}, s j) * s a)) = v * s a := by
      rw [← mul_sum, ← sum_mul, hμ, one_mul]
    rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat, if_pos rfl,
      ← huv, ← hμ', sum_congr rfl]
    intro x hx
    rw [mul_assoc, if_neg fun ha : x = a ↦ hat (ha.casesOn hx)]
    rw [mul_assoc]
    congr
    

Proof state:
case h.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
h✝ : Finset.Nonempty t✝
a : I
t : Finset I
hat : a ∉ t
h : Finset.Nonempty t
ih : (∃ μ, ∑ i in t, μ i * ∏ j in t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i in t, μ i * ∏ j in t \ {i}, s j = 1
u v : R
huv : u * ∏ i in t, s i + v * s a = 1
hμ' : ∑ i in t, v * ((μ i * ∏ j in t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ ∏ j in insert a t \ {x}, s j = (∏ j in t \ {x}, s j) * s a

Next tactic:
rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]