/-- The convergents coincide in the expected way at the squashed position if the partial denominator
at the squashed position is not zero. -/
theorem succ_nth_convergent_eq_squashGCF_nth_convergent [Field K]
    (nth_part_denom_ne_zero : ∀ {b : K}, g.partialDenominators.get? n = some b → b ≠ 0) :
    g.convergents (n + 1) = (squashGCF g n).convergents n := by
  cases' Decidable.em (g.TerminatedAt n) with terminated_at_n not_terminated_at_n
  · have : squashGCF g n = g := squashGCF_eq_self_of_terminated terminated_at_n
    simp only [this, convergents_stable_of_terminated n.le_succ terminated_at_n]
  · obtain ⟨⟨a, b⟩, s_nth_eq⟩ : ∃ gp_n, g.s.get? n = some gp_n
    exact Option.ne_none_iff_exists'.mp not_terminated_at_n
    have b_ne_zero : b ≠ 0 := nth_part_denom_ne_zero (part_denom_eq_s_b s_nth_eq)
    cases' n with n'
    case zero =>
      suffices (b * g.h + a) / b = g.h + a / b by
        simpa [squashGCF, s_nth_eq, convergent_eq_conts_a_div_conts_b,
          continuants_recurrenceAux s_nth_eq zeroth_continuant_aux_eq_one_zero
            first_continuant_aux_eq_h_one]
      calc
        (b * g.h + a) / b = b * g.h / b + a / b := by ring
        -- requires `Field`, not `DivisionRing`
        _ = g.h + a / b := by rw [mul_div_cancel_left _ b_ne_zero]
    case succ =>
      obtain ⟨⟨pa, pb⟩, s_n'th_eq⟩ : ∃ gp_n', g.s.get? n' = some gp_n' :=
        g.s.ge_stable n'.le_succ s_nth_eq
      -- Notations
      let g' := squashGCF g (n' + 1)
      set pred_conts := g.continuantsAux (n' + 1) with succ_n'th_conts_aux_eq
      set ppred_conts := g.continuantsAux n' with n'th_conts_aux_eq
      let pA := pred_conts.a
      let pB := pred_conts.b
      let ppA := ppred_conts.a
      let ppB := ppred_conts.b
      set pred_conts' := g'.continuantsAux (n' + 1) with succ_n'th_conts_aux_eq'
      set ppred_conts' := g'.continuantsAux n' with n'th_conts_aux_eq'
      let pA' := pred_conts'.a
      let pB' := pred_conts'.b
      let ppA' := ppred_conts'.a
      let ppB' := ppred_conts'.b
      -- first compute the convergent of the squashed gcf
      have : g'.convergents (n' + 1) =
          ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB') := by
        have : g'.s.get? n' = some ⟨pa, pb + a / b⟩ :=
          squashSeq_nth_of_not_terminated s_n'th_eq s_nth_eq
        rw [convergent_eq_conts_a_div_conts_b,
          continuants_recurrenceAux this n'th_conts_aux_eq'.symm succ_n'th_conts_aux_eq'.symm]
      rw [this]
      -- then compute the convergent of the original gcf by recursively unfolding the continuants
      -- computation twice
      have : g.convergents (n' + 2) =
          (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) := by
        -- use the recurrence once
        have : g.continuantsAux (n' + 2) = ⟨pb * pA + pa * ppA, pb * pB + pa * ppB⟩ :=
          continuantsAux_recurrence s_n'th_eq n'th_conts_aux_eq.symm succ_n'th_conts_aux_eq.symm
        -- and a second time
        rw [convergent_eq_conts_a_div_conts_b,
          continuants_recurrenceAux s_nth_eq succ_n'th_conts_aux_eq.symm this]
      rw [this]
      suffices
        ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =
          (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) by
        obtain ⟨eq1, eq2, eq3, eq4⟩ : pA' = pA ∧ pB' = pB ∧ ppA' = ppA ∧ ppB' = ppB := by
          simp [*, (continuantsAux_eq_continuantsAux_squashGCF_of_le <| le_refl <| n' + 1).symm,
            (continuantsAux_eq_continuantsAux_squashGCF_of_le n'.le_succ).symm]
        symm
        simpa only [eq1, eq2, eq3, eq4, mul_div_cancel _ b_ne_zero]
      

Proof state:
case intro.mk
K : Type u_1
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : Field K
a b : K
b_ne_zero : b ≠ 0
n' : ℕ
nth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0
not_terminated_at_n : ¬TerminatedAt g (Nat.succ n')
s_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }
pa pb : K
s_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }
g' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)
pred_conts : Pair K := continuantsAux g (n' + 1)
succ_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)
ppred_conts : Pair K := continuantsAux g n'
n'th_conts_aux_eq : ppred_conts = continuantsAux g n'
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
pred_conts' : Pair K := continuantsAux g' (n' + 1)
succ_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)
ppred_conts' : Pair K := continuantsAux g' n'
n'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'
pA' : K := pred_conts'.a
pB' : K := pred_conts'.b
ppA' : K := ppred_conts'.a
ppB' : K := ppred_conts'.b
this✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')
this : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)
⊢ ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =
    (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)

Next tactic:
field_simp