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algebra.continued_fractions.computation.terminates_iff_rat

# Termination of Continued Fraction Computations (gcf.of) #

## Summary #

We show that the continued fraction for a value v, as defined in algebra.continued_fractions.computation.basic, terminates if and only if v corresponds to a rational number, that is ↑v = q for some q : ℚ.

## Main Theorems #

• gcf.coe_of_rat shows that gcf.of v = gcf.of q for v : α given that ↑v = q and q : ℚ.
• gcf.terminates_iff_rat shows that gcf.of v terminates if and only if ↑v = q for some q : ℚ.

## Tags #

rational, continued fraction, termination

We want to show that the computation of a continued fraction gcf.of v terminates if and only if v ∈ ℚ. In the next section, we show the implication from left to right.

We first show that every finite convergent corresponds to a rational number q and then use the finite correctness proof (of_correctness_of_terminates) of gcf.of to show that v = ↑q.

theorem generalized_continued_fraction.exists_rat_eq_nth_numerator {K : Type u_1} [floor_ring K] (v : K) (n : ) :
∃ (q : ),
theorem generalized_continued_fraction.exists_rat_eq_nth_denominator {K : Type u_1} [floor_ring K] (v : K) (n : ) :
∃ (q : ),
theorem generalized_continued_fraction.exists_rat_eq_nth_convergent {K : Type u_1} [floor_ring K] (v : K) (n : ) :
∃ (q : ),

Every finite convergent corresponds to a rational number.

theorem generalized_continued_fraction.exists_rat_eq_of_terminates {K : Type u_1} [floor_ring K] {v : K} (terminates : .terminates) :
∃ (q : ), v = q

Every terminating continued fraction corresponds to a rational number.

Before we can show that the continued fraction of a rational number terminates, we have to prove some technical translation lemmas. More precisely, in this section, we show that, given a rational number q : ℚ and value v : K with v = ↑q, the continued fraction of q and v coincide. In particular, we show that (↑(gcf.of q : gcf ℚ) : gcf K) = gcf.of v in gcf.coe_of_rat.

To do this, we proceed bottom-up, showing the correspondence between the basic functions involved in the computation first and then lift the results step-by-step.

First, we show the correspondence for the very basic functions in gcf.int_fract_pair.

theorem generalized_continued_fraction.int_fract_pair.coe_of_rat_eq {K : Type u_1} [floor_ring K] [archimedean K] {v : K} {q : } (v_eq_q : v = q) :
theorem generalized_continued_fraction.int_fract_pair.coe_stream_nth_rat_eq {K : Type u_1} [floor_ring K] [archimedean K] {v : K} {q : } (v_eq_q : v = q) (n : ) :
theorem generalized_continued_fraction.int_fract_pair.coe_stream_rat_eq {K : Type u_1} [floor_ring K] [archimedean K] {v : K} {q : } (v_eq_q : v = q) :

Now we lift the coercion results to the continued fraction computation.

theorem generalized_continued_fraction.coe_of_h_rat_eq {K : Type u_1} [floor_ring K] [archimedean K] {v : K} {q : } (v_eq_q : v = q) :
theorem generalized_continued_fraction.coe_of_s_nth_rat_eq {K : Type u_1} [floor_ring K] [archimedean K] {v : K} {q : } (v_eq_q : v = q) (n : ) :
theorem generalized_continued_fraction.coe_of_s_rat_eq {K : Type u_1} [floor_ring K] [archimedean K] {v : K} {q : } (v_eq_q : v = q) :
theorem generalized_continued_fraction.coe_of_rat_eq {K : Type u_1} [floor_ring K] [archimedean K] {v : K} {q : } (v_eq_q : v = q) :

Given (v : K), (q : ℚ), and v = q, we have that gcf.of q = gcf.of v

theorem generalized_continued_fraction.of_terminates_iff_of_rat_terminates {K : Type u_1} [floor_ring K] [archimedean K] {v : K} {q : } (v_eq_q : v = q) :

Finally, we show that the continued fraction of a rational number terminates.

The crucial insight is that, given any q : ℚ with 0 < q < 1, the numerator of fract q is smaller than the numerator of q. As the continued fraction computation recursively operates on the fractional part of a value v and 0 ≤ fract v < 1, we infer that the numerator of the fractional part in the computation decreases by at least one in each step. As 0 ≤ fract v, this process must stop after finite number of steps, and the computation hence terminates.

Shows that for any q : ℚ with 0 < q < 1, the numerator of the fractional part of int_fract_pair.of q⁻¹ is smaller than the numerator of q.

theorem generalized_continued_fraction.int_fract_pair.stream_succ_nth_fr_num_lt_nth_fr_num_rat {q : } {n : } {ifp_n ifp_succ_n : generalized_continued_fraction.int_fract_pair } (stream_nth_eq : = some ifp_n) (stream_succ_nth_eq : = some ifp_succ_n) :
ifp_succ_n.fr.num < ifp_n.fr.num

Shows that the sequence of numerators of the fractional parts of the stream is strictly monotonically decreasing.

The continued fraction of a rational number terminates.

theorem generalized_continued_fraction.terminates_iff_rat {K : Type u_1} [floor_ring K] [archimedean K] (v : K) :
∃ (q : ), v = q

The continued fraction gcf.of v terminates if and only if v ∈ ℚ.