awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
⊢ HAdd.hAdd (α := K) (↑(fib n) : K) (↑(fib (n + 1)) : K) ≤ gp.a * ppconts.b + gp.b * pconts.b
have succ : HAdd.hAdd (α := K) (↑(fib n) : K) (↑(fib (n + 1)) : K) ≤ ppconts.b + gp.b * pconts.b
Tactic: suffices (fib n : K) + fib (n + 1) ≤ ppconts.b + gp.b * pconts.b by
          simpa [of_partNum_eq_one <| partNum_eq_s_a s_ppred_nth_eq]
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
⊢ HAdd.hAdd (α := K) (↑(fib n) : K) (↑(fib (n + 1)) : K) ≤ ppconts.b + gp.b * pconts.b
have not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
Tactic: have not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1) :=
          mt (terminated_stable <| Nat.sub_le n 1) not_terminatedAt_n
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
⊢ HAdd.hAdd (α := K) (↑(fib n) : K) (↑(fib (n + 1)) : K) ≤ ppconts.b + gp.b * pconts.b
have not_terminatedAt_ppred_n : ¬g.TerminatedAt (n - 2)
Tactic: have not_terminatedAt_ppred_n : ¬TerminatedAt g (n - 2) :=
          mt (terminated_stable (n - 1).pred_le) not_terminatedAt_pred_n
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
not_terminatedAt_ppred_n : ¬g.TerminatedAt (n - 2)
⊢ HAdd.hAdd (α := K) (↑(fib n) : K) (↑(fib (n + 1)) : K) ≤ ppconts.b + gp.b * pconts.b
have ppred_nth_fib_le_ppconts_B : (↑(fib n) : K) ≤ ppconts.b
Tactic: have ppred_nth_fib_le_ppconts_B : (fib n : K) ≤ ppconts.b :=
          IH n (lt_trans (Nat.lt_add_one n) <| Nat.lt_add_one <| n + 1)
            (Or.inr not_terminatedAt_ppred_n)
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
not_terminatedAt_ppred_n : ¬g.TerminatedAt (n - 2)
ppred_nth_fib_le_ppconts_B : (↑(fib n) : K) ≤ ppconts.b
⊢ HAdd.hAdd (α := K) (↑(fib n) : K) (↑(fib (n + 1)) : K) ≤ ppconts.b + gp.b * pconts.b
have succ : (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b
Tactic: suffices (fib (n + 1) : K) ≤ gp.b * pconts.b by gcongr
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this_1 : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
not_terminatedAt_ppred_n : ¬g.TerminatedAt (n - 2)
ppred_nth_fib_le_ppconts_B : (↑(fib n) : K) ≤ ppconts.b
this : HMul.hMul (α := K) 1 (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b
⊢ (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b
have rwa : ∀ (this : (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b), (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b
Tactic: rwa [one_mul] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
not_terminatedAt_ppred_n : ¬g.TerminatedAt (n - 2)
ppred_nth_fib_le_ppconts_B : (↑(fib n) : K) ≤ ppconts.b
⊢ (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b
have succ : HMul.hMul (α := K) 1 (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b
Tactic: suffices 1 * (fib (n + 1) : K) ≤ gp.b * pconts.b by rwa [one_mul] at this
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
not_terminatedAt_ppred_n : ¬g.TerminatedAt (n - 2)
ppred_nth_fib_le_ppconts_B : (↑(fib n) : K) ≤ ppconts.b
⊢ HMul.hMul (α := K) 1 (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b
have one_le_gp_b : 1 ≤ gp.b
Tactic: have one_le_gp_b : (1 : K) ≤ gp.b :=
          of_one_le_get?_partDen (partDen_eq_s_b s_ppred_nth_eq)
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
n : ℕ
IH : ∀ m < n + 1 + 1, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → (↑(fib m) : K) ≤ ((of v).contsAux m).b
hyp : n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 + 1 - 2)
g : GenContFract K := of v
this : ¬n + 2 ≤ 1
not_terminatedAt_n : ¬g.TerminatedAt n
gp : Pair K
s_ppred_nth_eq : g.s.get? n = some gp
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
ppconts_eq : ppconts = g.contsAux n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
not_terminatedAt_ppred_n : ¬g.TerminatedAt (n - 2)
ppred_nth_fib_le_ppconts_B : (↑(fib n) : K) ≤ ppconts.b
one_le_gp_b : 1 ≤ gp.b
⊢ HMul.hMul (α := K) 1 (↑(fib (n + 1)) : K) ≤ gp.b * pconts.b
have succ : (↑(fib (n + 1)) : K) ≤ pconts.b
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
hyp : n = 0 ∨ ¬(of v).TerminatedAt (n - 1)
⊢ (↑(fib (n + 1)) : K) ≤ (of v).dens n
have rw : (↑(fib (n + 1)) : K) ≤ ((of v).contsAux (n + 1)).b
Tactic: rw [den_eq_conts_b, nth_cont_eq_succ_nth_contAux]
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
hyp : n = 0 ∨ ¬(of v).TerminatedAt (n - 1)
⊢ n + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n - 1)
have zero : ∀ (hyp : Eq (α := ℕ) 0 0 ∨ ¬(of v).TerminatedAt (HSub.hSub (α := ℕ) 0 1)),
  HAdd.hAdd (α := ℕ) 0 1 ≤ 1 ∨ ¬(of v).TerminatedAt (HSub.hSub (α := ℕ) 0 1)
Tactic: cases n with
    | zero => exact Or.inl <| le_refl 1
    | succ n => exact Or.inr (Or.resolve_left hyp n.succ_ne_zero)
Cleared hypotheses: n, hyp
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
hyp : n = 0 ∨ ¬(of v).TerminatedAt (n - 1)
zero :
  ∀ (hyp : Eq (α := ℕ) 0 0 ∨ ¬(of v).TerminatedAt (HSub.hSub (α := ℕ) 0 1)),
    HAdd.hAdd (α := ℕ) 0 1 ≤ 1 ∨ ¬(of v).TerminatedAt (HSub.hSub (α := ℕ) 0 1)
⊢ n + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n - 1)
have succ : ∀ (n : ℕ) (hyp : n + 1 = 0 ∨ ¬(of v).TerminatedAt (n + 1 - 1)), n + 1 + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n + 1 - 1)
Tactic: cases n with
    | zero => exact Or.inl <| le_refl 1
    | succ n => exact Or.inr (Or.resolve_left hyp n.succ_ne_zero)
Cleared hypotheses: n, hyp
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
hyp : n = 0 ∨ ¬(of v).TerminatedAt (n - 1)
⊢ (↑(fib (n + 1)) : K) ≤ ((of v).contsAux (n + 1)).b
have this : n + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n - 1)
Tactic: have : n + 1 ≤ 1 ∨ ¬(of v).TerminatedAt (n - 1) := by
    cases n with
    | zero => exact Or.inl <| le_refl 1
    | succ n => exact Or.inr (Or.resolve_left hyp n.succ_ne_zero)
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux n).b
⊢ 0 ≤ ((of v).contsAux (n + 1)).b
have succ : ∀ (terminated : g.TerminatedAt (n - 1)), 0 ≤ ((of v).contsAux (n + 1)).b
Tactic: rcases Decidable.em <| g.TerminatedAt (n - 1) with terminated | not_terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux n).b
succ : ∀ (terminated : g.TerminatedAt (n - 1)), 0 ≤ ((of v).contsAux (n + 1)).b
⊢ 0 ≤ ((of v).contsAux (n + 1)).b
have succ₁ : ∀ (not_terminated : ¬g.TerminatedAt (n - 1)), 0 ≤ ((of v).contsAux (n + 1)).b
Tactic: rcases Decidable.em <| g.TerminatedAt (n - 1) with terminated | not_terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux n).b
terminated : g.TerminatedAt (n - 1)
⊢ 0 ≤ ((of v).contsAux (n + 1)).b
have succ : ∀ (IH : 0 ≤ ((of v).contsAux 0).b) (terminated : g.TerminatedAt (HSub.hSub (α := ℕ) 0 1)),
  0 ≤ ((of v).contsAux (HAdd.hAdd (α := ℕ) 0 1)).b
Tactic: rcases n with - | n
Cleared hypotheses: n, IH, terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux n).b
terminated : g.TerminatedAt (n - 1)
succ :
  ∀ (IH : 0 ≤ ((of v).contsAux 0).b) (terminated : g.TerminatedAt (HSub.hSub (α := ℕ) 0 1)),
    0 ≤ ((of v).contsAux (HAdd.hAdd (α := ℕ) 0 1)).b
⊢ 0 ≤ ((of v).contsAux (n + 1)).b
have succ₁ : ∀ (n : ℕ) (IH : 0 ≤ ((of v).contsAux (n + 1)).b) (terminated : g.TerminatedAt (n + 1 - 1)),
  0 ≤ ((of v).contsAux (n + 1 + 1)).b
Tactic: rcases n with - | n
Cleared hypotheses: n, IH, terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n_1 : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux (n + 1)).b
terminated : g.TerminatedAt (n + 1 - 1)
⊢ 0 ≤ ((of v).contsAux (n + 1 + 1)).b
have this : g.contsAux (n + 2) = g.contsAux (n + 1)
Tactic: have : g.contsAux (n + 2) = g.contsAux (n + 1) :=
          contsAux_stable_step_of_terminated terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
⊢ 0 ≤ ((of v).contsAux n).b
have zero : LE.le.{u_1} (α := K) 0 0
Tactic: induction n with
  | zero => rfl
  | succ n IH =>
    rcases Decidable.em <| g.TerminatedAt (n - 1) with terminated | not_terminated
    · -- terminating case
      rcases n with - | n
      · simp [zero_le_one]
      · have : g.contsAux (n + 2) = g.contsAux (n + 1) :=
          contsAux_stable_step_of_terminated terminated
        simp only [g, this, IH]
    · -- non-terminating case
      calc
        (0 : K) ≤ fib (n + 1) := mod_cast (n + 1).fib.zero_le
        _ ≤ ((of v).contsAux (n + 1)).b := fib_le_of_contsAux_b (Or.inr not_terminated)
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
zero : LE.le.{u_1} (α := K) 0 0
⊢ 0 ≤ ((of v).contsAux n).b
have succ : ∀ (n : ℕ) (IH : 0 ≤ ((of v).contsAux n).b), 0 ≤ ((of v).contsAux (n + 1)).b
Tactic: induction n with
  | zero => rfl
  | succ n IH =>
    rcases Decidable.em <| g.TerminatedAt (n - 1) with terminated | not_terminated
    · -- terminating case
      rcases n with - | n
      · simp [zero_le_one]
      · have : g.contsAux (n + 2) = g.contsAux (n + 1) :=
          contsAux_stable_step_of_terminated terminated
        simp only [g, this, IH]
    · -- non-terminating case
      calc
        (0 : K) ≤ fib (n + 1) := mod_cast (n + 1).fib.zero_le
        _ ≤ ((of v).contsAux (n + 1)).b := fib_le_of_contsAux_b (Or.inr not_terminated)
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
⊢ 0 ≤ (of v).dens n
have rw : 0 ≤ ((of v).contsAux (n + 1)).b
Tactic: rw [den_eq_conts_b, nth_cont_eq_succ_nth_contAux]
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
b : K
nth_partDen_eq : (of v).partDens.get? n = some b
⊢ b * ((of v).contsAux (n + 1)).b ≤ ((of v).contsAux (n + 2)).b
have obtain : ∀ (gp_n : Pair K) (nth_s_eq : (of v).s.get? n = some gp_n) (nth_partDen_eq : (of v).partDens.get? n = some gp_n.b),
  gp_n.b * ((of v).contsAux (n + 1)).b ≤ ((of v).contsAux (n + 2)).b
Tactic: obtain ⟨gp_n, nth_s_eq, rfl⟩ : ∃ gp_n, (of v).s.get? n = some gp_n ∧ gp_n.b = b :=
    exists_s_b_of_partDen nth_partDen_eq
Cleared hypotheses: b, nth_partDen_eq
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
b : K
nth_partDenom_eq : (of v).partDens.get? n = some b
⊢ b * (of v).dens n ≤ (of v).dens (n + 1)
have rw : b * ((of v).contsAux (n + 1)).b ≤ (of v).dens (n + 1)
Tactic: rw [den_eq_conts_b, nth_cont_eq_succ_nth_contAux]
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have inl : ∀ (terminated : g.partDens.TerminatedAt n), (of v).dens n ≤ (of v).dens (n + 1)
Tactic: rcases Decidable.em <| g.partDens.TerminatedAt n with terminated | not_terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
inl : ∀ (terminated : g.partDens.TerminatedAt n), (of v).dens n ≤ (of v).dens (n + 1)
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have inr : ∀ (not_terminated : ¬g.partDens.TerminatedAt n), (of v).dens n ≤ (of v).dens (n + 1)
Tactic: rcases Decidable.em <| g.partDens.TerminatedAt n with terminated | not_terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
terminated : g.partDens.TerminatedAt n
⊢ g.partDens.get? n = none
have rwa : ∀ (terminated : g.partDens.get? n = none), g.partDens.get? n = none
Tactic: rwa [Stream'.Seq.TerminatedAt] at terminated
Cleared hypotheses: terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
terminated : g.partDens.TerminatedAt n
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have this : g.partDens.get? n = none
Tactic: have : g.partDens.get? n = none := by rwa [Stream'.Seq.TerminatedAt] at terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
terminated : g.partDens.TerminatedAt n
this : g.partDens.get? n = none
⊢ g.partDens.get? n = none
have rwa : ∀ (terminated this : g.partDens.get? n = none), g.partDens.get? n = none
Tactic: rwa [Stream'.Seq.TerminatedAt] at terminated
Cleared hypotheses: terminated, this
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
terminated : g.partDens.TerminatedAt n
this : g.partDens.get? n = none
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have this₁ : g.TerminatedAt n
Tactic: have : g.TerminatedAt n :=
      terminatedAt_iff_partDen_none.2 (by rwa [Stream'.Seq.TerminatedAt] at terminated)
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
terminated : g.partDens.TerminatedAt n
this_1 : g.partDens.get? n = none
this : g.TerminatedAt n
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have this₁ : g.dens (n + 1) = g.dens n
Tactic: have : g.dens (n + 1) = g.dens n :=
      dens_stable_of_terminated n.le_succ this
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
terminated : g.partDens.TerminatedAt n
this_1 : g.partDens.get? n = none
this_2 : g.TerminatedAt n
this : g.dens (n + 1) = g.dens n
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have inl : (of v).dens n ≤ (of v).dens n
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
not_terminated : ¬g.partDens.TerminatedAt n
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have inr : ∀ (b : K) (nth_partDen_eq : g.partDens.get? n = some b), (of v).dens n ≤ (of v).dens (n + 1)
Tactic: obtain ⟨b, nth_partDen_eq⟩ : ∃ b, g.partDens.get? n = some b :=
      Option.ne_none_iff_exists'.mp not_terminated
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
not_terminated : ¬g.partDens.TerminatedAt n
b : K
nth_partDen_eq : g.partDens.get? n = some b
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have this : 1 ≤ b
Tactic: have : 1 ≤ b := of_one_le_get?_partDen nth_partDen_eq
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (goal)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
g : GenContFract K := of v
not_terminated : ¬g.partDens.TerminatedAt n
b : K
nth_partDen_eq : g.partDens.get? n = some b
this : 1 ≤ b
⊢ (of v).dens n ≤ (of v).dens (n + 1)
have n₁ : g.dens n ≤ b * g.dens n
Tactic: calc
      g.dens n ≤ b * g.dens n := by
        simpa using mul_le_mul_of_nonneg_right this zero_le_of_den
      _ ≤ g.dens (n + 1) := le_of_succ_get?_den nth_partDen_eq
-- Lean 4 proof step from Mathlib.Algebra.ContinuedFractions.Computation.Approximations (hypothesis)
Goal:
K : Type u_1
v : K
n : ℕ
inst : Field K
inst_1 : LinearOrder K
inst_2 : IsStrictOrderedRing K
inst_3 : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else HPow.hPow (α := K) (-1) n / (B * (ifp.fr⁻¹ * B + pB))
have g_finite_correctness : v = GenContFract.compExactValue pred_conts conts ifp.fr
Tactic: have g_finite_correctness :
    v = GenContFract.compExactValue pred_conts conts ifp.fr :=
    compExactValue_correctness_of_stream_eq_some stream_nth_eq