# mathlib3documentation

algebra.continued_fractions.translations

# Basic Translation Lemmas Between Functions Defined for Continued Fractions #

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## Summary #

Some simple translation lemmas between the different definitions of functions defined in algebra.continued_fractions.basic.

### Translations Between General Access Functions #

Here we give some basic translations that hold by definition between the various methods that allow us to access the numerators and denominators of a continued fraction.

theorem generalized_continued_fraction.part_num_eq_s_a {α : Type u_1} {n : } (s_nth_eq : g.s.nth n = ) :
theorem generalized_continued_fraction.part_denom_eq_s_b {α : Type u_1} {n : } (s_nth_eq : g.s.nth n = ) :
theorem generalized_continued_fraction.exists_s_a_of_part_num {α : Type u_1} {n : } {a : α} (nth_part_num_eq : = ) :
(gp : , g.s.nth n = gp.a = a
theorem generalized_continued_fraction.exists_s_b_of_part_denom {α : Type u_1} {n : } {b : α} (nth_part_denom_eq : = ) :
(gp : , g.s.nth n = gp.b = b

### Translations Between Computational Functions #

Here we give some basic translations that hold by definition for the computational methods of a continued fraction.

theorem generalized_continued_fraction.exists_conts_a_of_num {K : Type u_1} {n : } {A : K} (nth_num_eq : g.numerators n = A) :
(conts : , g.continuants n = conts conts.a = A
theorem generalized_continued_fraction.exists_conts_b_of_denom {K : Type u_1} {n : } {B : K} (nth_denom_eq : g.denominators n = B) :
(conts : , g.continuants n = conts conts.b = B
@[simp]
theorem generalized_continued_fraction.second_continuant_aux_eq {K : Type u_1} (zeroth_s_eq : g.s.nth 0 = ) :
= {a := gp.b * g.h + gp.a, b := gp.b}
theorem generalized_continued_fraction.first_continuant_eq {K : Type u_1} (zeroth_s_eq : g.s.nth 0 = ) :
g.continuants 1 = {a := gp.b * g.h + gp.a, b := gp.b}
theorem generalized_continued_fraction.first_numerator_eq {K : Type u_1} (zeroth_s_eq : g.s.nth 0 = ) :
g.numerators 1 = gp.b * g.h + gp.a
theorem generalized_continued_fraction.first_denominator_eq {K : Type u_1} (zeroth_s_eq : g.s.nth 0 = ) :
@[simp]