Termination of Continued Fraction Computations (GeneralizedContinuedFraction.of)

Summary

We show that the continued fraction for a value v, as defined in Mathlib.Algebra.ContinuedFractions.Basic, terminates if and only if v corresponds to a rational number, that is ↑v = q for some q : ℚ.

Main Theorems

• GeneralizedContinuedFraction.coe_of_rat_eq shows that GeneralizedContinuedFraction.of v = GeneralizedContinuedFraction.of q for v : α given that ↑v = q and q : ℚ.
• GeneralizedContinuedFraction.terminates_iff_rat shows that GeneralizedContinuedFraction.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 GeneralizedContinuedFraction.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 GeneralizedContinuedFraction.of to show that v = ↑q.

theorem GeneralizedContinuedFraction.exists_gcf_pair_rat_eq_of_nth_conts_aux {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) (n : ℕ) : ∃ (conts : GeneralizedContinuedFraction.Pair ℚ), GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) n = GeneralizedContinuedFraction.Pair.map Rat.cast conts

theorem GeneralizedContinuedFraction.exists_gcf_pair_rat_eq_nth_conts {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) (n : ℕ) : ∃ (conts : GeneralizedContinuedFraction.Pair ℚ), GeneralizedContinuedFraction.continuants (GeneralizedContinuedFraction.of v) n = GeneralizedContinuedFraction.Pair.map Rat.cast conts

theorem GeneralizedContinuedFraction.exists_rat_eq_nth_numerator {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) (n : ℕ) : ∃ (q : ℚ), GeneralizedContinuedFraction.numerators (GeneralizedContinuedFraction.of v) n = ↑q

theorem GeneralizedContinuedFraction.exists_rat_eq_nth_denominator {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) (n : ℕ) : ∃ (q : ℚ), GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n = ↑q

theorem GeneralizedContinuedFraction.exists_rat_eq_nth_convergent {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) (n : ℕ) : ∃ (q : ℚ), GeneralizedContinuedFraction.convergents (GeneralizedContinuedFraction.of v) n = ↑q

Every finite convergent corresponds to a rational number.

theorem GeneralizedContinuedFraction.exists_rat_eq_of_terminates {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} (terminates : GeneralizedContinuedFraction.Terminates (GeneralizedContinuedFraction.of v)) : ∃ (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

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

in GeneralizedContinuedFraction.coe_of_rat_eq.

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 GeneralizedContinuedFraction.IntFractPair.

theorem GeneralizedContinuedFraction.IntFractPair.coe_of_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) : GeneralizedContinuedFraction.IntFractPair.mapFr Rat.cast (GeneralizedContinuedFraction.IntFractPair.of q) = GeneralizedContinuedFraction.IntFractPair.of v

theorem GeneralizedContinuedFraction.IntFractPair.coe_stream_nth_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) (n : ℕ) : Option.map (GeneralizedContinuedFraction.IntFractPair.mapFr Rat.cast) (GeneralizedContinuedFraction.IntFractPair.stream q n) = GeneralizedContinuedFraction.IntFractPair.stream v n

theorem GeneralizedContinuedFraction.IntFractPair.coe_stream'_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) : Stream'.map (Option.map (GeneralizedContinuedFraction.IntFractPair.mapFr Rat.cast)) (GeneralizedContinuedFraction.IntFractPair.stream q) = GeneralizedContinuedFraction.IntFractPair.stream v

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

theorem GeneralizedContinuedFraction.coe_of_h_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) : ↑(GeneralizedContinuedFraction.of q).h = (GeneralizedContinuedFraction.of v).h

theorem GeneralizedContinuedFraction.coe_of_s_get?_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) (n : ℕ) : Option.map (GeneralizedContinuedFraction.Pair.map Rat.cast) (Stream'.Seq.get? (GeneralizedContinuedFraction.of q).s n) = Stream'.Seq.get? (GeneralizedContinuedFraction.of v).s n

theorem GeneralizedContinuedFraction.coe_of_s_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) : Stream'.Seq.map (GeneralizedContinuedFraction.Pair.map Rat.cast) (GeneralizedContinuedFraction.of q).s = (GeneralizedContinuedFraction.of v).s

theorem GeneralizedContinuedFraction.coe_of_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) : { h := ↑(GeneralizedContinuedFraction.of q).h, s := Stream'.Seq.map (GeneralizedContinuedFraction.Pair.map Rat.cast) (GeneralizedContinuedFraction.of q).s } = GeneralizedContinuedFraction.of v

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

theorem GeneralizedContinuedFraction.of_terminates_iff_of_rat_terminates {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : ℚ} (v_eq_q : v = ↑q) : GeneralizedContinuedFraction.Terminates (GeneralizedContinuedFraction.of v) ↔ GeneralizedContinuedFraction.Terminates (GeneralizedContinuedFraction.of q)

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 GeneralizedContinuedFraction.IntFractPair.of_inv_fr_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) : (GeneralizedContinuedFraction.IntFractPair.of q⁻¹).fr.num < q.num

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

theorem GeneralizedContinuedFraction.IntFractPair.stream_succ_nth_fr_num_lt_nth_fr_num_rat {q : ℚ} {n : ℕ} {ifp_n : GeneralizedContinuedFraction.IntFractPair ℚ} {ifp_succ_n : GeneralizedContinuedFraction.IntFractPair ℚ} (stream_nth_eq : GeneralizedContinuedFraction.IntFractPair.stream q n = some ifp_n) (stream_succ_nth_eq : GeneralizedContinuedFraction.IntFractPair.stream q (n + 1) = 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 GeneralizedContinuedFraction.IntFractPair.stream_nth_fr_num_le_fr_num_sub_n_rat {q : ℚ} {n : ℕ} {ifp_n : GeneralizedContinuedFraction.IntFractPair ℚ} : GeneralizedContinuedFraction.IntFractPair.stream q n = some ifp_n → ifp_n.fr.num ≤ (GeneralizedContinuedFraction.IntFractPair.of q).fr.num - ↑n

theorem GeneralizedContinuedFraction.IntFractPair.exists_nth_stream_eq_none_of_rat (q : ℚ) : ∃ (n : ℕ), GeneralizedContinuedFraction.IntFractPair.stream q n = none

theorem GeneralizedContinuedFraction.terminates_of_rat (q : ℚ) : GeneralizedContinuedFraction.Terminates (GeneralizedContinuedFraction.of q)

The continued fraction of a rational number terminates.

theorem GeneralizedContinuedFraction.terminates_iff_rat {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) : GeneralizedContinuedFraction.Terminates (GeneralizedContinuedFraction.of v) ↔ ∃ (q : ℚ), v = ↑q

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