/-- Shows the correctness of `compExactValue` in case the continued fraction
`GeneralizedContinuedFraction.of v` did not terminate at position `n`. That is, we obtain the
value `v` if we pass the two successive (auxiliary) continuants at positions `n` and `n + 1` as well
as the fractional part at `IntFractPair.stream n` to `compExactValue`.

The correctness might be seen more readily if one uses `convergents'` to evaluate the continued
fraction. Here is an example to illustrate the idea:

Let `(v : ℚ) := 3.4`. We have
- `GeneralizedContinuedFraction.IntFractPair.stream v 0 = some ⟨3, 0.4⟩`, and
- `GeneralizedContinuedFraction.IntFractPair.stream v 1 = some ⟨2, 0.5⟩`.
Now `(GeneralizedContinuedFraction.of v).convergents' 1 = 3 + 1/2`, and our fractional term at
position `2` is `0.5`. We hence have `v = 3 + 1/(2 + 0.5) = 3 + 1/2.5 = 3.4`. This computation
corresponds exactly to the one using the recurrence equation in `compExactValue`.
-/
theorem compExactValue_correctness_of_stream_eq_some :
    ∀ {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n →
      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux <| n + 1) ifp_n.fr := by
  let g := of v
  induction' n with n IH
  · intro ifp_zero stream_zero_eq
    -- Nat.zero
    have : IntFractPair.of v = ifp_zero := by
      have : IntFractPair.stream v 0 = some (IntFractPair.of v) := rfl
      simpa only [Nat.zero_eq, this, Option.some.injEq] using stream_zero_eq
    cases this
    cases' Decidable.em (Int.fract v = 0) with fract_eq_zero fract_ne_zero
    -- Int.fract v = 0; we must then have `v = ⌊v⌋`
    · suffices v = ⌊v⌋ by
        -- Porting note: was `simpa [continuantsAux, fract_eq_zero, compExactValue]`
        field_simp [nextContinuants, nextNumerator, nextDenominator, compExactValue]
        have : (IntFractPair.of v).fr = Int.fract v := rfl
        rwa [this, if_pos fract_eq_zero]
      calc
        v = Int.fract v + ⌊v⌋ := by rw [Int.fract_add_floor]
        _ = ⌊v⌋ := by simp [fract_eq_zero]
    -- Int.fract v ≠ 0; the claim then easily follows by unfolding a single computation step
    · field_simp [continuantsAux, nextContinuants, nextNumerator, nextDenominator,
        of_h_eq_floor, compExactValue]
      -- Porting note: this and the if_neg rewrite are needed
      have : (IntFractPair.of v).fr = Int.fract v := rfl
      rw [this, if_neg fract_ne_zero, Int.floor_add_fract]
  · intro ifp_succ_n succ_nth_stream_eq
    -- Nat.succ
    obtain ⟨ifp_n, nth_stream_eq, nth_fract_ne_zero, -⟩ :
      ∃ 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
inst✝¹ : LinearOrderedField K
v : K
n✝ : ℕ
inst✝ : FloorRing K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH :
  ∀ {ifp_n : IntFractPair K},
    IntFractPair.stream v n = some ifp_n →
      v = compExactValue (continuantsAux (of v) n) (continuantsAux (of v) (n + 1)) ifp_n.fr
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (Nat.succ n) = some 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
case succ.intro.intro.intro
K : Type u_1
inst✝¹ : LinearOrderedField K
v : K
n✝ : ℕ
inst✝ : FloorRing K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH :
  ∀ {ifp_n : IntFractPair K},
    IntFractPair.stream v n = some ifp_n →
      v = compExactValue (continuantsAux (of v) n) (continuantsAux (of v) (n + 1)) ifp_n.fr
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (Nat.succ n) = some ifp_succ_n
ifp_n : IntFractPair K
nth_stream_eq : IntFractPair.stream v n = some ifp_n
nth_fract_ne_zero : ifp_n.fr ≠ 0
⊢ v = compExactValue (continuantsAux (of v) (Nat.succ n)) (continuantsAux (of v) (Nat.succ n + 1)) ifp_succ_n.fr

Next tactic:
exact IntFractPair.succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq