Euclidean algorithm for ℕ #

This file sets up a version of the Euclidean algorithm that only works with natural numbers. Given 0 < a, b, it computes the unique (w, x, y, z, d) such that the following identities hold:

This story is closely related to the structure of SL₂(ℕ) (as a free monoid on two generators) and the theory of continued fractions.

Main declarations #

Notes #

See Nat.Xgcd for a very similar algorithm allowing values in .

structure PNat.XgcdType :
  • wp is a variable which changes through the algorithm.

    wp :
  • x satisfies a / d = w + x at the final step.

    x :
  • y satisfies b / d = z + y at the final step.

    y :
  • zp is a variable which changes through the algorithm.

    zp :
  • ap is a variable which changes through the algorithm.

    ap :
  • bp is a variable which changes through the algorithm.

    bp :

A term of XgcdType is a system of six naturals. They should be thought of as representing the matrix [[w, x], [y, z]] = [[wp + 1, x], [y, zp + 1]] together with the vector [a, b] = [ap + 1, bp + 1].

Instances For

    The Repr instance converts terms to strings in a way that reflects the matrix/vector interpretation as above.

    • One or more equations did not get rendered due to their size.
    def' (w : ℕ+) (x : ) (y : ) (z : ℕ+) (a : ℕ+) (b : ℕ+) :

    Another mk using ℕ and ℕ+


    r = a % b: remainder


    q = ap / bp: quotient


    The map v gives the product of the matrix [[w, x], [y, z]] = [[wp + 1, x], [y, zp + 1]] and the vector [a, b] = [ap + 1, bp + 1]. The map vp gives [sp, tp] such that v = [sp + 1, tp + 1].


    v = [sp + 1, tp + 1], check vp


    succ₂ [t.1, t.2] = [t.1.succ, t.2.succ]


    IsSpecial holds if the matrix has determinant one.


    IsReduced holds if the two entries in the vector are the same. The reduction algorithm will produce a system with this property, whose product vector is the same as for the original system.


    flip flips the placement of variables during the algorithm.


    Properties of division with remainder for a / b.

    The following function provides the starting point for our algorithm. We will apply an iterative reduction process to it, which will produce a system satisfying IsReduced. The gcd can be read off from this final system.


    finish happens when the reducing process ends.


    This is the main reduction step, which is used when u.r ≠ 0, or equivalently b does not divide a.


    We will apply the above step recursively. The following result is used to ensure that the process terminates.

    The reduction step does not change the product vector.

    We can now define the full reduction function, which applies step as long as possible, and then applies finish. Note that the "have" statement puts a fact in the local context, and the equation compiler uses this fact to help construct the full definition in terms of well-founded recursion. The same fact needs to be introduced in all the inductive proofs of properties given below.

    def PNat.xgcd (a : ℕ+) (b : ℕ+) :

    Extended Euclidean algorithm

    def PNat.gcdD (a : ℕ+) (b : ℕ+) :

    gcdD a b = gcd a b

    def PNat.gcdW (a : ℕ+) (b : ℕ+) :

    Final value of w

    def PNat.gcdX (a : ℕ+) (b : ℕ+) :

    Final value of x

    def PNat.gcdY (a : ℕ+) (b : ℕ+) :

    Final value of y

    def PNat.gcdZ (a : ℕ+) (b : ℕ+) :

    Final value of z

    def PNat.gcdA' (a : ℕ+) (b : ℕ+) :

    Final value of a / d

    def PNat.gcdB' (a : ℕ+) (b : ℕ+) :

    Final value of b / d

    theorem PNat.gcdA'_coe (a : ℕ+) (b : ℕ+) :
    ↑(PNat.gcdA' a b) = ↑(PNat.gcdW a b) + PNat.gcdX a b
    theorem PNat.gcdB'_coe (a : ℕ+) (b : ℕ+) :
    ↑(PNat.gcdB' a b) = PNat.gcdY a b + ↑(PNat.gcdZ a b)
    theorem PNat.gcd_props (a : ℕ+) (b : ℕ+) :
    let d := PNat.gcdD a b; let w := PNat.gcdW a b; let x := PNat.gcdX a b; let y := PNat.gcdY a b; let z := PNat.gcdZ a b; let a' := PNat.gcdA' a b; let b' := PNat.gcdB' a b; w * z = Nat.succPNat (x * y) a = a' * d b = b' * d z * a' = Nat.succPNat (x * b') w * b' = Nat.succPNat (y * a') z * a = x * b + d w * b = y * a + d
    theorem PNat.gcd_eq (a : ℕ+) (b : ℕ+) :
    theorem PNat.gcd_a_eq (a : ℕ+) (b : ℕ+) :
    theorem PNat.gcd_b_eq (a : ℕ+) (b : ℕ+) :
    theorem PNat.gcd_rel_left (a : ℕ+) (b : ℕ+) :
    ↑(PNat.gcdZ a b) * a = PNat.gcdX a b * b + ↑(PNat.gcd a b)
    theorem PNat.gcd_rel_right (a : ℕ+) (b : ℕ+) :
    ↑(PNat.gcdW a b) * b = PNat.gcdY a b * a + ↑(PNat.gcd a b)