K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp ppred pred : Pair K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_conts_eq : continuantsAux g (n + 1) = ppred
succ_nth_conts_eq : continuantsAux g (n + 1 + 1) = pred
⊢ continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp ppred pred : Pair K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_conts_eq : continuants g n = ppred
succ_nth_conts_eq : continuants g (n + 1) = pred
⊢ continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }
rw [nth_cont_eq_succ_nth_cont_aux] at nth_conts_eq succ_nth_conts_eq
no goals
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp ppred pred : Pair K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_conts_eq : continuantsAux g (n + 1) = ppred
succ_nth_conts_eq : continuantsAux g (n + 1 + 1) = pred
⊢ continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }
exact continuants_recurrenceAux succ_nth_s_eq nth_conts_eq succ_nth_conts_eq