/-- Shows that `|v - Aₙ / Bₙ| ≤ 1 / (Bₙ * Bₙ₊₁)`. -/
theorem abs_sub_convergents_le (not_terminated_at_n : ¬(of v).TerminatedAt n) :
    |v - (of v).convergents n| ≤ 1 / ((of v).denominators n * ((of v).denominators <| n + 1)) := by
  -- shorthand notation
  let g := of v
  let nextConts := g.continuantsAux (n + 2)
  set conts := continuantsAux g (n + 1) with conts_eq
  set pred_conts := continuantsAux g n with pred_conts_eq
  -- change the goal to something more readable
  change |v - convergents g n| ≤ 1 / (conts.b * nextConts.b)
  obtain ⟨gp, s_nth_eq⟩ : ∃ gp, g.s.get? n = some gp
  exact Option.ne_none_iff_exists'.1 not_terminated_at_n
  have gp_a_eq_one : gp.a = 1 := of_part_num_eq_one (part_num_eq_s_a s_nth_eq)
  -- unfold the recurrence relation for `nextConts.b`
  have nextConts_b_eq : nextConts.b = pred_conts.b + gp.b * conts.b := by
    simp [continuantsAux_recurrence s_nth_eq pred_conts_eq conts_eq, gp_a_eq_one,
      pred_conts_eq.symm, conts_eq.symm, add_comm]
  let denom := conts.b * (pred_conts.b + gp.b * conts.b)
  suffices |v - g.convergents n| ≤ 1 / denom by rw [nextConts_b_eq]; congr 1
  obtain ⟨ifp_succ_n, succ_nth_stream_eq, ifp_succ_n_b_eq_gp_b⟩ :
    ∃ ifp_succ_n, IntFractPair.stream v (n + 1) = some ifp_succ_n ∧ (ifp_succ_n.b : K) = gp.b
  exact IntFractPair.exists_succ_get?_stream_of_gcf_of_get?_eq_some s_nth_eq
  obtain ⟨ifp_n, stream_nth_eq, stream_nth_fr_ne_zero, if_of_eq_ifp_succ_n⟩ :
    ∃ ifp_n,
      IntFractPair.stream v n = some ifp_n ∧ ifp_n.fr ≠ 0 ∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n
  

Proof state:
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
not_terminated_at_n : ¬TerminatedAt (of v) n
g : GeneralizedContinuedFraction K := of v
nextConts : Pair K := continuantsAux g (n + 2)
conts : Pair K := continuantsAux g (n + 1)
conts_eq : conts = continuantsAux g (n + 1)
pred_conts : Pair K := continuantsAux g n
pred_conts_eq : pred_conts = continuantsAux g n
gp : Pair K
s_nth_eq : Stream'.Seq.get? g.s n = some gp
gp_a_eq_one : gp.a = 1
nextConts_b_eq : nextConts.b = pred_conts.b + gp.b * conts.b
denom : K := conts.b * (pred_conts.b + gp.b * conts.b)
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n
ifp_succ_n_b_eq_gp_b : ↑ifp_succ_n.b = gp.b
⊢ ∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧ ifp_n.fr ≠ 0 ∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n
case intro.intro.intro.intro.intro.intro
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
not_terminated_at_n : ¬TerminatedAt (of v) n
g : GeneralizedContinuedFraction K := of v
nextConts : Pair K := continuantsAux g (n + 2)
conts : Pair K := continuantsAux g (n + 1)
conts_eq : conts = continuantsAux g (n + 1)
pred_conts : Pair K := continuantsAux g n
pred_conts_eq : pred_conts = continuantsAux g n
gp : Pair K
s_nth_eq : Stream'.Seq.get? g.s n = some gp
gp_a_eq_one : gp.a = 1
nextConts_b_eq : nextConts.b = pred_conts.b + gp.b * conts.b
denom : K := conts.b * (pred_conts.b + gp.b * conts.b)
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n
ifp_succ_n_b_eq_gp_b : ↑ifp_succ_n.b = gp.b
ifp_n : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp_n
stream_nth_fr_ne_zero : ifp_n.fr ≠ 0
if_of_eq_ifp_succ_n : IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n
⊢ |v - convergents g n| ≤ 1 / denom

Next tactic:
exact IntFractPair.succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq