Correctness of Terminating Continued Fraction Computations (GeneralizedContinuedFraction.of)

Summary

We show the correctness of the algorithm computing continued fractions (GeneralizedContinuedFraction.of) in case of termination in the following sense:

At every step n : ℕ, we can obtain the value v by adding a specific residual term to the last denominator of the fraction described by (GeneralizedContinuedFraction.of v).convergents' n. The residual term will be zero exactly when the continued fraction terminated; otherwise, the residual term will be given by the fractional part stored in GeneralizedContinuedFraction.IntFractPair.stream v n.

For an example, refer to GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some and for more information about the computation process, refer to Algebra.ContinuedFractions.Computation.Basic.

Main definitions

• GeneralizedContinuedFraction.compExactValue can be used to compute the exact value approximated by the continued fraction GeneralizedContinuedFraction.of v by adding a residual term as described in the summary.

Main Theorems

• GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some shows that GeneralizedContinuedFraction.compExactValue indeed returns the value v when given the convergent and fractional part as described in the summary.
• GeneralizedContinuedFraction.of_correctness_of_terminatedAt shows the equality v = (GeneralizedContinuedFraction.of v).convergents n if GeneralizedContinuedFraction.of v terminated at position n.

def GeneralizedContinuedFraction.compExactValue {K : Type u_1} [LinearOrderedField K] (pconts : GeneralizedContinuedFraction.Pair K) (conts : GeneralizedContinuedFraction.Pair K) (fr : K) : K

Given two continuants pconts and conts and a value fr, this function returns

• conts.a / conts.b if fr = 0
• exact_conts.a / exact_conts.b where exact_conts = nextContinuants 1 fr⁻¹ pconts conts otherwise.

This function can be used to compute the exact value approximated by a continued fraction GeneralizedContinuedFraction.of v as described in lemma compExactValue_correctness_of_stream_eq_some.

Equations

• One or more equations did not get rendered due to their size.

theorem GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some_aux_comp {K : Type u_1} [LinearOrderedField K] [FloorRing K] {a : K} (b : K) (c : K) (fract_a_ne_zero : Int.fract a ≠ 0) : (↑⌊a⌋ * b + c) / Int.fract a + b = (b * a + c) / Int.fract a

Just a computational lemma we need for the next main proof.

theorem GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some {K : Type u_1} [LinearOrderedField K] {v : K} {n : ℕ} [FloorRing K] {ifp_n : GeneralizedContinuedFraction.IntFractPair K} : GeneralizedContinuedFraction.IntFractPair.stream v n = some ifp_n → v = GeneralizedContinuedFraction.compExactValue (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) n) (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) (n + 1)) ifp_n.fr

Shows the correctness of compExactValue in case the continued fraction GeneralizedContinuedFraction.of v did not terminate at position n. That is, we obtain the value v if we pass the two successive (auxiliary) continuants at positions n and n + 1 as well as the fractional part at IntFractPair.stream n to compExactValue.

The correctness might be seen more readily if one uses convergents' to evaluate the continued fraction. Here is an example to illustrate the idea:

Let (v : ℚ) := 3.4. We have

• GeneralizedContinuedFraction.IntFractPair.stream v 0 = some ⟨3, 0.4⟩, and
• GeneralizedContinuedFraction.IntFractPair.stream v 1 = some ⟨2, 0.5⟩. Now (GeneralizedContinuedFraction.of v).convergents' 1 = 3 + 1/2, and our fractional term at position 2 is 0.5. We hence have v = 3 + 1/(2 + 0.5) = 3 + 1/2.5 = 3.4. This computation corresponds exactly to the one using the recurrence equation in compExactValue.

theorem GeneralizedContinuedFraction.of_correctness_of_nth_stream_eq_none {K : Type u_1} [LinearOrderedField K] {v : K} {n : ℕ} [FloorRing K] (nth_stream_eq_none : GeneralizedContinuedFraction.IntFractPair.stream v n = none) : v = GeneralizedContinuedFraction.convergents (GeneralizedContinuedFraction.of v) (n - 1)

The convergent of GeneralizedContinuedFraction.of v at step n - 1 is exactly v if the IntFractPair.stream of the corresponding continued fraction terminated at step n.

theorem GeneralizedContinuedFraction.of_correctness_of_terminatedAt {K : Type u_1} [LinearOrderedField K] {v : K} {n : ℕ} [FloorRing K] (terminated_at_n : GeneralizedContinuedFraction.TerminatedAt (GeneralizedContinuedFraction.of v) n) : v = GeneralizedContinuedFraction.convergents (GeneralizedContinuedFraction.of v) n

If GeneralizedContinuedFraction.of v terminated at step n, then the nth convergent is exactly v.

theorem GeneralizedContinuedFraction.of_correctness_of_terminates {K : Type u_1} [LinearOrderedField K] {v : K} [FloorRing K] (terminates : GeneralizedContinuedFraction.Terminates (GeneralizedContinuedFraction.of v)) : ∃ (n : ℕ), v = GeneralizedContinuedFraction.convergents (GeneralizedContinuedFraction.of v) n

If GeneralizedContinuedFraction.of v terminates, then there is n : ℕ such that the nth convergent is exactly v.

theorem GeneralizedContinuedFraction.of_correctness_atTop_of_terminates {K : Type u_1} [LinearOrderedField K] {v : K} [FloorRing K] (terminates : GeneralizedContinuedFraction.Terminates (GeneralizedContinuedFraction.of v)) : ∀ᶠ (n : ℕ) in Filter.atTop, v = GeneralizedContinuedFraction.convergents (GeneralizedContinuedFraction.of v) n

If GeneralizedContinuedFraction.of v terminates, then its convergents will eventually always be v.