/-- 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
  exact IntFractPair.succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq
  let denom' := conts.b * (pred_conts.b + ifp_n.fr⁻¹ * conts.b)
  -- now we can use `sub_convergents_eq` to simplify our goal
  suffices |(-1) ^ n / denom'| ≤ 1 / denom by
    have : v - g.convergents n = (-1) ^ n / denom' := by
      -- apply `sub_convergents_eq` and simplify the result
      have tmp := sub_convergents_eq stream_nth_eq
      simp only [stream_nth_fr_ne_zero, conts_eq.symm, pred_conts_eq.symm, if_false] at tmp
      rw [tmp]
      ring
    rwa [this]
  -- derive some tedious inequalities that we need to rewrite our goal
  have nextConts_b_ineq : (fib (n + 2) : K) ≤ pred_conts.b + gp.b * conts.b := by
    have : (fib (n + 2) : K) ≤ nextConts.b :=
      fib_le_of_continuantsAux_b (Or.inr not_terminated_at_n)
    rwa [nextConts_b_eq] at this
  have conts_b_ineq : (fib (n + 1) : K) ≤ conts.b :=
    haveI : ¬g.TerminatedAt (n - 1) := mt (terminated_stable n.pred_le) not_terminated_at_n
    fib_le_of_continuantsAux_b <| Or.inr this
  have zero_lt_conts_b : 0 < conts.b :=
    conts_b_ineq.trans_lt' $ mod_cast fib_pos.2 n.succ_pos
  -- `denom'` is positive, so we can remove `|⬝|` from our goal
  suffices 1 / denom' ≤ 1 / denom by
    have : |(-1) ^ n / denom'| = 1 / denom' := by
      suffices 1 / |denom'| = 1 / denom' by rwa [abs_div, abs_neg_one_pow n]
      have : 0 < denom' := by
        have : 0 ≤ pred_conts.b :=
          haveI : (fib n : K) ≤ pred_conts.b :=
            haveI : ¬g.TerminatedAt (n - 2) :=
              mt (terminated_stable (n.sub_le 2)) not_terminated_at_n
            fib_le_of_continuantsAux_b <| Or.inr this
          le_trans (mod_cast (fib n).zero_le) this
        have : 0 < ifp_n.fr⁻¹ :=
          haveI zero_le_ifp_n_fract : 0 ≤ ifp_n.fr :=
            IntFractPair.nth_stream_fr_nonneg stream_nth_eq
          inv_pos.2 (lt_of_le_of_ne zero_le_ifp_n_fract stream_nth_fr_ne_zero.symm)
        -- Porting note: replaced complicated positivity proof with tactic.
        positivity
      rw [abs_of_pos this]
    rwa [this]
  suffices 0 < denom ∧ denom ≤ denom' from div_le_div_of_le_left zero_le_one this.left this.right
  constructor
  · have : 0 < pred_conts.b + gp.b * conts.b :=
      nextConts_b_ineq.trans_lt' $ mod_cast fib_pos.2 $ succ_pos _
    solve_by_elim [mul_pos]
  · -- we can cancel multiplication by `conts.b` and addition with `pred_conts.b`
    suffices : gp.b * conts.b ≤ ifp_n.fr⁻¹ * conts.b
    exact (mul_le_mul_left zero_lt_conts_b).2 <| (add_le_add_iff_left pred_conts.b).2 this
    suffices (ifp_succ_n.b : K) * conts.b ≤ ifp_n.fr⁻¹ * conts.b by rwa [← ifp_succ_n_b_eq_gp_b]
    

Proof state:
case this
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
denom' : K := conts.b * (pred_conts.b + ifp_n.fr⁻¹ * conts.b)
nextConts_b_ineq : ↑(fib (n + 2)) ≤ pred_conts.b + gp.b * conts.b
conts_b_ineq : ↑(fib (n + 1)) ≤ conts.b
zero_lt_conts_b : 0 < conts.b
⊢ ↑ifp_succ_n.b * conts.b ≤ ifp_n.fr⁻¹ * conts.b

Next tactic:
have : (ifp_succ_n.b : K) ≤ ifp_n.fr⁻¹ :=
      IntFractPair.succ_nth_stream_b_le_nth_stream_fr_inv stream_nth_eq succ_nth_stream_eq