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algebra.continued_fractions.computation.terminates_iff_rat

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

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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 #

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

Tags #

rational, continued fraction, termination

Terminating Continued Fractions Are Rational #

We want to show that the computation of a continued fraction generalized_continued_fraction.of v terminates if and only if v ∈ ℚ. In this 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 generalized_continued_fraction.of to show that v = ↑q.

theorem generalized_continued_fraction.exists_gcf_pair_rat_eq_of_nth_conts_aux {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) (n : ℕ) :

∃ (conts : generalized_continued_fraction.pair ℚ), (generalized_continued_fraction.of v).continuants_aux n = generalized_continued_fraction.pair.map coe conts

theorem generalized_continued_fraction.exists_gcf_pair_rat_eq_nth_conts {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) (n : ℕ) :

∃ (conts : generalized_continued_fraction.pair ℚ), (generalized_continued_fraction.of v).continuants n = generalized_continued_fraction.pair.map coe conts

theorem generalized_continued_fraction.exists_rat_eq_nth_numerator {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) (n : ℕ) :

∃ (q : ℚ), (generalized_continued_fraction.of v).numerators n = ↑q

theorem generalized_continued_fraction.exists_rat_eq_nth_denominator {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) (n : ℕ) :

∃ (q : ℚ), (generalized_continued_fraction.of v).denominators n = ↑q

theorem generalized_continued_fraction.exists_rat_eq_nth_convergent {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) (n : ℕ) :

∃ (q : ℚ), (generalized_continued_fraction.of v).convergents n = ↑q

Every finite convergent corresponds to a rational number.

theorem generalized_continued_fraction.exists_rat_eq_of_terminates {K : Type u_1} [linear_ordered_field K] [floor_ring K] {v : K} (terminates : (generalized_continued_fraction.of v).terminates) :

∃ (q : ℚ), v = ↑q

Every terminating continued fraction corresponds to a rational number.

Technical Translation Lemmas #

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

    (↑(generalized_continued_fraction.of q : generalized_continued_fraction ℚ)
      : generalized_continued_fraction K)
  = generalized_continued_fraction.of v`

in generalized_continued_fraction.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 generalized_continued_fraction.int_fract_pair.

theorem generalized_continued_fraction.int_fract_pair.coe_of_rat_eq {K : Type u_1} [linear_ordered_field K] [floor_ring K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) :

generalized_continued_fraction.int_fract_pair.mapFr coe (generalized_continued_fraction.int_fract_pair.of q) = generalized_continued_fraction.int_fract_pair.of v

theorem generalized_continued_fraction.int_fract_pair.coe_stream_nth_rat_eq {K : Type u_1} [linear_ordered_field K] [floor_ring K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) (n : ℕ) :

option.map (generalized_continued_fraction.int_fract_pair.mapFr coe) (generalized_continued_fraction.int_fract_pair.stream q n) = generalized_continued_fraction.int_fract_pair.stream v n

theorem generalized_continued_fraction.int_fract_pair.coe_stream_rat_eq {K : Type u_1} [linear_ordered_field K] [floor_ring K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) :

stream.map (option.map (generalized_continued_fraction.int_fract_pair.mapFr coe)) (generalized_continued_fraction.int_fract_pair.stream q) = generalized_continued_fraction.int_fract_pair.stream v

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

theorem generalized_continued_fraction.coe_of_h_rat_eq {K : Type u_1} [linear_ordered_field K] [floor_ring K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) :

↑((generalized_continued_fraction.of q).h) = (generalized_continued_fraction.of v).h

theorem generalized_continued_fraction.coe_of_s_nth_rat_eq {K : Type u_1} [linear_ordered_field K] [floor_ring K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) (n : ℕ) :

option.map (generalized_continued_fraction.pair.map coe) ((generalized_continued_fraction.of q).s.nth n) = (generalized_continued_fraction.of v).s.nth n

theorem generalized_continued_fraction.coe_of_s_rat_eq {K : Type u_1} [linear_ordered_field K] [floor_ring K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) :

stream.seq.map (generalized_continued_fraction.pair.map coe) (generalized_continued_fraction.of q).s = (generalized_continued_fraction.of v).s

theorem generalized_continued_fraction.coe_of_rat_eq {K : Type u_1} [linear_ordered_field K] [floor_ring K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) :

{h := ↑((generalized_continued_fraction.of q).h), s := stream.seq.map (generalized_continued_fraction.pair.map coe) (generalized_continued_fraction.of q).s} = generalized_continued_fraction.of v

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} [linear_ordered_field K] [floor_ring K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) :

(generalized_continued_fraction.of v).terminates ↔ (generalized_continued_fraction.of q).terminates

Continued Fractions of Rationals Terminate #

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 int.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 ≤ int.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 ≤ int.fract v, this process must stop after finite number of steps, and the computation hence terminates.

theorem generalized_continued_fraction.int_fract_pair.of_inv_fr_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) :

(generalized_continued_fraction.int_fract_pair.of q⁻¹).fr.num < q.num

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 : ℚ} {n : ℕ} {ifp_n ifp_succ_n : generalized_continued_fraction.int_fract_pair ℚ} (stream_nth_eq : generalized_continued_fraction.int_fract_pair.stream q n = option.some ifp_n) (stream_succ_nth_eq : generalized_continued_fraction.int_fract_pair.stream q (n + 1) = option.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 antitone.

theorem generalized_continued_fraction.int_fract_pair.stream_nth_fr_num_le_fr_num_sub_n_rat {q : ℚ} {n : ℕ} {ifp_n : generalized_continued_fraction.int_fract_pair ℚ} :

generalized_continued_fraction.int_fract_pair.stream q n = option.some ifp_n → ifp_n.fr.num ≤ (generalized_continued_fraction.int_fract_pair.of q).fr.num - ↑n

theorem generalized_continued_fraction.int_fract_pair.exists_nth_stream_eq_none_of_rat (q : ℚ) :

∃ (n : ℕ), generalized_continued_fraction.int_fract_pair.stream q n = option.none

theorem generalized_continued_fraction.terminates_of_rat (q : ℚ) :

(generalized_continued_fraction.of q).terminates

The continued fraction of a rational number terminates.

theorem generalized_continued_fraction.terminates_iff_rat {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) :

(generalized_continued_fraction.of v).terminates ↔ ∃ (q : ℚ), v = ↑q

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