/-- This lemma follows from the finite correctness proof, the determinant equality, and
by simplifying the difference. -/
theorem sub_convergents_eq {ifp : IntFractPair K}
    (stream_nth_eq : IntFractPair.stream v n = some ifp) :
    let g := of v
    let B := (g.continuantsAux (n + 1)).b
    let pB := (g.continuantsAux n).b
    v - g.convergents n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB)) := by
  -- set up some shorthand notation
  let g := of v
  let conts := g.continuantsAux (n + 1)
  let pred_conts := g.continuantsAux n
  have g_finite_correctness :
    v = GeneralizedContinuedFraction.compExactValue pred_conts conts ifp.fr :=
    compExactValue_correctness_of_stream_eq_some stream_nth_eq
  obtain (ifp_fr_eq_zero | ifp_fr_ne_zero) := eq_or_ne ifp.fr 0
  · suffices v - g.convergents n = 0 by simpa [ifp_fr_eq_zero]
    replace g_finite_correctness : v = g.convergents n;
    · simpa [GeneralizedContinuedFraction.compExactValue, ifp_fr_eq_zero] using g_finite_correctness
    exact sub_eq_zero.2 g_finite_correctness
  · -- more shorthand notation
    let A := conts.a
    let B := conts.b
    let pA := pred_conts.a
    let pB := pred_conts.b
    -- first, let's simplify the goal as `ifp.fr ≠ 0`
    suffices v - A / B = (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB)) by simpa [ifp_fr_ne_zero]
    -- now we can unfold `g.compExactValue` to derive the following equality for `v`
    replace g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
    · simpa [GeneralizedContinuedFraction.compExactValue, ifp_fr_ne_zero, nextContinuants,
        nextNumerator, nextDenominator, add_comm] using g_finite_correctness
    -- let's rewrite this equality for `v` in our goal
    suffices
      (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B) - A / B = (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB)) by
      rwa [g_finite_correctness]
    -- To continue, we need use the determinant equality. So let's derive the needed hypothesis.
    have n_eq_zero_or_not_terminated_at_pred_n : n = 0 ∨ ¬g.TerminatedAt (n - 1) := by
      cases' n with n'
      · simp
      · have : IntFractPair.stream v (n' + 1) ≠ none := by simp [stream_nth_eq]
        have : ¬g.TerminatedAt n' :=
          (not_congr of_terminatedAt_n_iff_succ_nth_intFractPair_stream_eq_none).2 this
        exact Or.inr this
    have determinant_eq : pA * B - pB * A = (-1) ^ n :=
      determinant_aux n_eq_zero_or_not_terminated_at_pred_n
    -- now all we got to do is to rewrite this equality in our goal and re-arrange terms;
    -- however, for this, we first have to derive quite a few tedious inequalities.
    have pB_ineq : (fib n : K) ≤ pB :=
      haveI : n ≤ 1 ∨ ¬g.TerminatedAt (n - 2) := by
        cases' n_eq_zero_or_not_terminated_at_pred_n with n_eq_zero not_terminated_at_pred_n
        · simp [n_eq_zero]
        · exact Or.inr <| mt (terminated_stable (n - 1).pred_le) not_terminated_at_pred_n
      fib_le_of_continuantsAux_b this
    have B_ineq : (fib (n + 1) : K) ≤ B :=
      haveI : n + 1 ≤ 1 ∨ ¬g.TerminatedAt (n + 1 - 2) := by
        cases' n_eq_zero_or_not_terminated_at_pred_n with n_eq_zero not_terminated_at_pred_n
        · simp [n_eq_zero, le_refl]
        · exact Or.inr not_terminated_at_pred_n
      fib_le_of_continuantsAux_b this
    have zero_lt_B : 0 < B := B_ineq.trans_lt' $ cast_pos.2 $ fib_pos.2 n.succ_pos
    have : 0 ≤ pB := (cast_nonneg _).trans pB_ineq
    have : 0 < ifp.fr :=
      ifp_fr_ne_zero.lt_of_le' $ IntFractPair.nth_stream_fr_nonneg stream_nth_eq
    

Proof state:
case inr
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GeneralizedContinuedFraction K := of v
conts : Pair K := continuantsAux g (n + 1)
pred_conts : Pair K := continuantsAux g n
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
n_eq_zero_or_not_terminated_at_pred_n : n = 0 ∨ ¬TerminatedAt g (n - 1)
determinant_eq : pA * B - pB * A = (-1) ^ n
pB_ineq : ↑(fib n) ≤ pB
B_ineq : ↑(fib (n + 1)) ≤ B
zero_lt_B : 0 < B
this✝ : 0 ≤ pB
this : 0 < ifp.fr
⊢ (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B) - A / B = (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))

Next tactic:
have : pB + ifp.fr⁻¹ * B ≠ 0 := by positivity