Module: Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv

Goal state before:
pp: K : Type u_1
n : ℕ
g : GenContFract K
inst✝ : Field K
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? n = some b → b ≠ 0
⊢ g.convs (n + 1) = (g.squashGCF n).convs n

Tactic:
by
  rcases Decidable.em (g.TerminatedAt n) with terminatedAt_n | not_terminatedAt_n
  · have : squashGCF g n = g := squashGCF_eq_self_of_terminated terminatedAt_n
    simp only [this, convs_stable_of_terminated n.le_succ terminatedAt_n]
  · obtain ⟨⟨a, b⟩, s_nth_eq⟩ : ∃ gp_n, g.s.get? n = some gp_n :=
      Option.ne_none_iff_exists'.mp not_terminatedAt_n
    have b_ne_zero : b ≠ 0 := nth_partDen_ne_zero (partDen_eq_s_b s_nth_eq)
    cases n with
    | zero =>
      suffices (b * g.h + a) / b = g.h + a / b by
        simpa [squashGCF, s_nth_eq, conv_eq_conts_a_div_conts_b,
          conts_recurrenceAux s_nth_eq zeroth_contAux_eq_one_zero first_contAux_eq_h_one]
      grind
    | succ n' =>
      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.contsAux (n' + 1) with succ_n'th_contsAux_eq
      set ppred_conts := g.contsAux n' with n'th_contsAux_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'.contsAux (n' + 1) with succ_n'th_contsAux_eq'
      set ppred_conts' := g'.contsAux n' with n'th_contsAux_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'.convs (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 [conv_eq_conts_a_div_conts_b,
          conts_recurrenceAux this n'th_contsAux_eq'.symm succ_n'th_contsAux_eq'.symm]
      rw [this]
      -- then compute the convergent of the original gcf by recursively unfolding the continuants
      -- computation twice
      have : g.convs (n' + 2) =
          (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) := by
        -- use the recurrence once
        have : g.contsAux (n' + 2) = ⟨pb * pA + pa * ppA, pb * pB + pa * ppB⟩ :=
          contsAux_recurrence s_n'th_eq n'th_contsAux_eq.symm succ_n'th_contsAux_eq.symm
        -- and a second time
        rw [conv_eq_conts_a_div_conts_b,
          conts_recurrenceAux s_nth_eq succ_n'th_contsAux_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 [*, g', pA, pB, ppA, ppB, pA', pB', ppA', ppB',
            (contsAux_eq_contsAux_squashGCF_of_le <| le_refl <| n' + 1).symm,
            (contsAux_eq_contsAux_squashGCF_of_le n'.le_succ).symm]
        symm
        simpa only [eq1, eq2, eq3, eq4, mul_div_cancel_right₀ _ b_ne_zero]
      grind

Goal state after: