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Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat

Termination of Continued Fraction Computations (GenContFract.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 #

Tags #

rational, continued fraction, termination

Terminating Continued Fractions Are Rational #

We want to show that the computation of a continued fraction GenContFract.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 GenContFract.of to show that v = ↑q.

theorem GenContFract.exists_rat_eq_nth_num {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) (n : ) :
∃ (q : ), (GenContFract.of v).nums n = q
theorem GenContFract.exists_rat_eq_nth_den {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) (n : ) :
∃ (q : ), (GenContFract.of v).dens n = q
theorem GenContFract.exists_rat_eq_nth_conv {K : Type u_1} [LinearOrderedField K] [FloorRing K] (v : K) (n : ) :
∃ (q : ), (GenContFract.of v).convs n = q

Every finite convergent corresponds to a rational number.

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

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

in GenContFract.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 GenContFract.IntFractPair.

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

theorem GenContFract.coe_of_h_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : } (v_eq_q : v = q) :
theorem GenContFract.coe_of_s_get?_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : } (v_eq_q : v = q) (n : ) :
Option.map (GenContFract.Pair.map Rat.cast) ((GenContFract.of q).s.get? n) = (GenContFract.of v).s.get? n
theorem GenContFract.coe_of_s_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : } (v_eq_q : v = q) :
theorem GenContFract.coe_of_rat_eq {K : Type u_1} [LinearOrderedField K] [FloorRing K] {v : K} {q : } (v_eq_q : v = q) :

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

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

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 GenContFract.IntFractPair.stream_succ_nth_fr_num_lt_nth_fr_num_rat {q : } {n : } {ifp_n : GenContFract.IntFractPair } {ifp_succ_n : GenContFract.IntFractPair } (stream_nth_eq : GenContFract.IntFractPair.stream q n = some ifp_n) (stream_succ_nth_eq : GenContFract.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 GenContFract.terminates_of_rat (q : ) :
(GenContFract.of q).Terminates

The continued fraction of a rational number terminates.

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

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