Library Coq.ZArith.Znumtheory


Require Import ZArith_base.
Require Import ZArithRing.
Require Import Zcomplements.
Require Import Zdiv.
Require Import Wf_nat.

For compatibility reasons, this Open Scope isn't local as it should

Open Scope Z_scope.

This file contains some notions of number theory upon Z numbers:
  • a divisibility predicate Z.divide
  • a gcd predicate gcd
  • Euclid algorithm euclid
  • a relatively prime predicate rel_prime
  • a prime predicate prime
  • properties of the efficient Z.gcd function

Notation Zgcd := Z.gcd (compat "8.3").
Notation Zggcd := Z.ggcd (compat "8.3").
Notation Zggcd_gcd := Z.ggcd_gcd (compat "8.3").
Notation Zggcd_correct_divisors := Z.ggcd_correct_divisors (compat "8.3").
Notation Zgcd_divide_l := Z.gcd_divide_l (compat "8.3").
Notation Zgcd_divide_r := Z.gcd_divide_r (compat "8.3").
Notation Zgcd_greatest := Z.gcd_greatest (compat "8.3").
Notation Zgcd_nonneg := Z.gcd_nonneg (compat "8.3").
Notation Zggcd_opp := Z.ggcd_opp (compat "8.3").

The former specialized inductive predicate Z.divide is now a generic existential predicate.

Notation Zdivide := Z.divide (compat "8.3").

Its former constructor is now a pseudo-constructor.

Definition Zdivide_intro a b q (H:b=q*a) : Z.divide a b := ex_intro _ q H.

Results concerning divisibility

Notation Zdivide_refl := Z.divide_refl (compat "8.3").
Notation Zone_divide := Z.divide_1_l (compat "8.3").
Notation Zdivide_0 := Z.divide_0_r (compat "8.3").
Notation Zmult_divide_compat_l := Z.mul_divide_mono_l (compat "8.3").
Notation Zmult_divide_compat_r := Z.mul_divide_mono_r (compat "8.3").
Notation Zdivide_plus_r := Z.divide_add_r (compat "8.3").
Notation Zdivide_minus_l := Z.divide_sub_r (compat "8.3").
Notation Zdivide_mult_l := Z.divide_mul_l (compat "8.3").
Notation Zdivide_mult_r := Z.divide_mul_r (compat "8.3").
Notation Zdivide_factor_r := Z.divide_factor_l (compat "8.3").
Notation Zdivide_factor_l := Z.divide_factor_r (compat "8.3").

Lemma Zdivide_opp_r a b : (a | b) -> (a | - b).

Lemma Zdivide_opp_r_rev a b : (a | - b) -> (a | b).

Lemma Zdivide_opp_l a b : (a | b) -> (- a | b).

Lemma Zdivide_opp_l_rev a b : (- a | b) -> (a | b).

Theorem Zdivide_Zabs_l a b : (Z.abs a | b) -> (a | b).

Theorem Zdivide_Zabs_inv_l a b : (a | b) -> (Z.abs a | b).

Hint Resolve Z.divide_refl Z.divide_1_l Z.divide_0_r: zarith.
Hint Resolve Z.mul_divide_mono_l Z.mul_divide_mono_r: zarith.
Hint Resolve Z.divide_add_r Zdivide_opp_r Zdivide_opp_r_rev Zdivide_opp_l
  Zdivide_opp_l_rev Z.divide_sub_r Z.divide_mul_l Z.divide_mul_r
  Z.divide_factor_l Z.divide_factor_r: zarith.

Auxiliary result.

Lemma Zmult_one x y : x >= 0 -> x * y = 1 -> x = 1.

Only 1 and -1 divide 1.

Notation Zdivide_1 := Z.divide_1_r (compat "8.3").

If a divides b and b divides a then a is b or -b.

Notation Zdivide_antisym := Z.divide_antisym (compat "8.3").
Notation Zdivide_trans := Z.divide_trans (compat "8.3").

If a divides b and b<>0 then |a| <= |b|.

Lemma Zdivide_bounds a b : (a | b) -> b <> 0 -> Z.abs a <= Z.abs b.

Z.divide can be expressed using Z.modulo.

Lemma Zmod_divide : forall a b, b<>0 -> a mod b = 0 -> (b | a).

Lemma Zdivide_mod : forall a b, (b | a) -> a mod b = 0.

Z.divide is hence decidable

Lemma Zdivide_dec a b : {(a | b)} + {~ (a | b)}.

Theorem Zdivide_Zdiv_eq a b : 0 < a -> (a | b) -> b = a * (b / a).

Theorem Zdivide_Zdiv_eq_2 a b c :
 0 < a -> (a | b) -> (c * b) / a = c * (b / a).

Theorem Zdivide_le: forall a b : Z,
 0 <= a -> 0 < b -> (a | b) -> a <= b.

Theorem Zdivide_Zdiv_lt_pos a b :
 1 < a -> 0 < b -> (a | b) -> 0 < b / a < b .

Lemma Zmod_div_mod n m a:
 0 < n -> 0 < m -> (n | m) -> a mod n = (a mod m) mod n.

Lemma Zmod_divide_minus a b c:
 0 < b -> a mod b = c -> (b | a - c).

Lemma Zdivide_mod_minus a b c:
 0 <= c < b -> (b | a - c) -> a mod b = c.

Greatest common divisor (gcd).

There is no unicity of the gcd; hence we define the predicate Zis_gcd a b g expressing that g is a gcd of a and b. (We show later that the gcd is actually unique if we discard its sign.)

Inductive Zis_gcd (a b g:Z) : Prop :=
 Zis_gcd_intro :
  (g | a) ->
  (g | b) ->
  (forall x, (x | a) -> (x | b) -> (x | g)) ->
  Zis_gcd a b g.

Trivial properties of gcd

Lemma Zis_gcd_sym : forall a b d, Zis_gcd a b d -> Zis_gcd b a d.

Lemma Zis_gcd_0 : forall a, Zis_gcd a 0 a.

Lemma Zis_gcd_1 : forall a, Zis_gcd a 1 1.

Lemma Zis_gcd_refl : forall a, Zis_gcd a a a.

Lemma Zis_gcd_minus : forall a b d, Zis_gcd a (- b) d -> Zis_gcd b a d.

Lemma Zis_gcd_opp : forall a b d, Zis_gcd a b d -> Zis_gcd b a (- d).

Lemma Zis_gcd_0_abs a : Zis_gcd 0 a (Z.abs a).

Hint Resolve Zis_gcd_sym Zis_gcd_0 Zis_gcd_minus Zis_gcd_opp: zarith.

Theorem Zis_gcd_unique: forall a b c d : Z,
 Zis_gcd a b c -> Zis_gcd a b d -> c = d \/ c = (- d).

Extended Euclid algorithm.

Euclid's algorithm to compute the gcd mainly relies on the following property.

Lemma Zis_gcd_for_euclid :
  forall a b d q:Z, Zis_gcd b (a - q * b) d -> Zis_gcd a b d.

Lemma Zis_gcd_for_euclid2 :
  forall b d q r:Z, Zis_gcd r b d -> Zis_gcd b (b * q + r) d.

We implement the extended version of Euclid's algorithm, i.e. the one computing Bezout's coefficients as it computes the gcd. We follow the algorithm given in Knuth's "Art of Computer Programming", vol 2, page 325.

Section extended_euclid_algorithm.

  Variables a b : Z.

The specification of Euclid's algorithm is the existence of u, v and d such that ua+vb=d and (gcd a b d).

  Inductive Euclid : Set :=
    Euclid_intro :
    forall u v d:Z, u * a + v * b = d -> Zis_gcd a b d -> Euclid.

The recursive part of Euclid's algorithm uses well-founded recursion of non-negative integers. It maintains 6 integers u1,u2,u3,v1,v2,v3 such that the following invariant holds: u1*a+u2*b=u3 and v1*a+v2*b=v3 and gcd(u3,v3)=gcd(a,b).

  Lemma euclid_rec :
    forall v3:Z,
      0 <= v3 ->
      forall u1 u2 u3 v1 v2:Z,
        u1 * a + u2 * b = u3 ->
        v1 * a + v2 * b = v3 ->
        (forall d:Z, Zis_gcd u3 v3 d -> Zis_gcd a b d) -> Euclid.

We get Euclid's algorithm by applying euclid_rec on 1,0,a,0,1,b when b>=0 and 1,0,a,0,-1,-b when b<0.

  Lemma euclid : Euclid.

End extended_euclid_algorithm.

Theorem Zis_gcd_uniqueness_apart_sign :
  forall a b d d':Z, Zis_gcd a b d -> Zis_gcd a b d' -> d = d' \/ d = - d'.

Bezout's coefficients


Inductive Bezout (a b d:Z) : Prop :=
  Bezout_intro : forall u v:Z, u * a + v * b = d -> Bezout a b d.

Existence of Bezout's coefficients for the gcd of a and b

Lemma Zis_gcd_bezout : forall a b d:Z, Zis_gcd a b d -> Bezout a b d.

gcd of ca and cb is c gcd(a,b).

Lemma Zis_gcd_mult :
  forall a b c d:Z, Zis_gcd a b d -> Zis_gcd (c * a) (c * b) (c * d).

Relative primality


Definition rel_prime (a b:Z) : Prop := Zis_gcd a b 1.

Bezout's theorem: a and b are relatively prime if and only if there exist u and v such that ua+vb = 1.

Lemma rel_prime_bezout : forall a b:Z, rel_prime a b -> Bezout a b 1.

Lemma bezout_rel_prime : forall a b:Z, Bezout a b 1 -> rel_prime a b.

Gauss's theorem: if a divides bc and if a and b are relatively prime, then a divides c.

Theorem Gauss : forall a b c:Z, (a | b * c) -> rel_prime a b -> (a | c).

If a is relatively prime to b and c, then it is to bc

Lemma rel_prime_mult :
  forall a b c:Z, rel_prime a b -> rel_prime a c -> rel_prime a (b * c).

Lemma rel_prime_cross_prod :
  forall a b c d:Z,
    rel_prime a b ->
    rel_prime c d -> b > 0 -> d > 0 -> a * d = b * c -> a = c /\ b = d.

After factorization by a gcd, the original numbers are relatively prime.

Lemma Zis_gcd_rel_prime :
  forall a b g:Z,
    b > 0 -> g >= 0 -> Zis_gcd a b g -> rel_prime (a / g) (b / g).

Theorem rel_prime_sym: forall a b, rel_prime a b -> rel_prime b a.

Theorem rel_prime_div: forall p q r,
 rel_prime p q -> (r | p) -> rel_prime r q.

Theorem rel_prime_1: forall n, rel_prime 1 n.

Theorem not_rel_prime_0: forall n, 1 < n -> ~ rel_prime 0 n.

Theorem rel_prime_mod: forall p q, 0 < q ->
 rel_prime p q -> rel_prime (p mod q) q.

Theorem rel_prime_mod_rev: forall p q, 0 < q ->
 rel_prime (p mod q) q -> rel_prime p q.

Theorem Zrel_prime_neq_mod_0: forall a b, 1 < b -> rel_prime a b -> a mod b <> 0.

Primality


Inductive prime (p:Z) : Prop :=
  prime_intro :
    1 < p -> (forall n:Z, 1 <= n < p -> rel_prime n p) -> prime p.

The sole divisors of a prime number p are -1, 1, p and -p.

Lemma prime_divisors :
  forall p:Z,
    prime p -> forall a:Z, (a | p) -> a = -1 \/ a = 1 \/ a = p \/ a = - p.

A prime number is relatively prime with any number it does not divide

Lemma prime_rel_prime :
  forall p:Z, prime p -> forall a:Z, ~ (p | a) -> rel_prime p a.

Hint Resolve prime_rel_prime: zarith.

As a consequence, a prime number is relatively prime with smaller numbers

Theorem rel_prime_le_prime:
 forall a p, prime p -> 1 <= a < p -> rel_prime a p.

If a prime p divides ab then it divides either a or b

Lemma prime_mult :
  forall p:Z, prime p -> forall a b:Z, (p | a * b) -> (p | a) \/ (p | b).

Lemma not_prime_0: ~ prime 0.

Lemma not_prime_1: ~ prime 1.

Lemma prime_2: prime 2.

Theorem prime_3: prime 3.

Theorem prime_ge_2 p : prime p -> 2 <= p.

Definition prime' p := 1<p /\ (forall n, 1<n<p -> ~ (n|p)).

Lemma Z_0_1_more x : 0<=x -> x=0 \/ x=1 \/ 1<x.

Theorem prime_alt p : prime' p <-> prime p.

Theorem square_not_prime: forall a, ~ prime (a * a).

Theorem prime_div_prime: forall p q,
 prime p -> prime q -> (p | q) -> p = q.

we now prove that Z.gcd is indeed a gcd in the sense of Zis_gcd.

Notation Zgcd_is_pos := Z.gcd_nonneg (compat "8.3").

Lemma Zgcd_is_gcd : forall a b, Zis_gcd a b (Z.gcd a b).

Theorem Zgcd_spec : forall x y : Z, {z : Z | Zis_gcd x y z /\ 0 <= z}.

Theorem Zdivide_Zgcd: forall p q r : Z,
 (p | q) -> (p | r) -> (p | Z.gcd q r).

Theorem Zis_gcd_gcd: forall a b c : Z,
 0 <= c -> Zis_gcd a b c -> Z.gcd a b = c.

Notation Zgcd_inv_0_l := Z.gcd_eq_0_l (compat "8.3").
Notation Zgcd_inv_0_r := Z.gcd_eq_0_r (compat "8.3").

Theorem Zgcd_div_swap0 : forall a b : Z,
 0 < Z.gcd a b ->
 0 < b ->
 (a / Z.gcd a b) * b = a * (b/Z.gcd a b).

Theorem Zgcd_div_swap : forall a b c : Z,
 0 < Z.gcd a b ->
 0 < b ->
 (c * a) / Z.gcd a b * b = c * a * (b/Z.gcd a b).

Notation Zgcd_comm := Z.gcd_comm (compat "8.3").

Lemma Zgcd_ass a b c : Z.gcd (Z.gcd a b) c = Z.gcd a (Z.gcd b c).

Notation Zgcd_Zabs := Z.gcd_abs_l (compat "8.3").
Notation Zgcd_0 := Z.gcd_0_r (compat "8.3").
Notation Zgcd_1 := Z.gcd_1_r (compat "8.3").

Hint Resolve Z.gcd_0_r Z.gcd_1_r : zarith.

Theorem Zgcd_1_rel_prime : forall a b,
 Z.gcd a b = 1 <-> rel_prime a b.

Definition rel_prime_dec: forall a b,
 { rel_prime a b }+{ ~ rel_prime a b }.

Definition prime_dec_aux:
 forall p m,
  { forall n, 1 < n < m -> rel_prime n p } +
  { exists n, 1 < n < m /\ ~ rel_prime n p }.

Definition prime_dec: forall p, { prime p }+{ ~ prime p }.

Theorem not_prime_divide:
 forall p, 1 < p -> ~ prime p -> exists n, 1 < n < p /\ (n | p).