Number theory

Parity properties

module Parity

  use int.Int

  predicate even (n: int) = exists k: int. n = 2 * k
  predicate odd (n: int) = exists k: int. n = 2 * k + 1

  lemma even_or_odd: forall n: int. even n \/ odd n

  lemma even_not_odd: forall n: int. even n -> not (odd n)
  lemma odd_not_even: forall n: int. odd n -> not (even n)

  lemma even_odd: forall n: int. even n -> odd (n + 1)
  lemma odd_even: forall n: int. odd n -> even (n + 1)

  lemma even_even: forall n: int. even n -> even (n + 2)
  lemma odd_odd: forall n: int. odd n -> odd (n + 2)

  lemma even_2k: forall k: int. even (2 * k)
  lemma odd_2k1: forall k: int. odd (2 * k + 1)

  use int.ComputerDivision

  lemma even_mod2 :
    forall n:int. even n <-> mod n 2 = 0

end

Divisibility

module Divisibility

  use export int.Int
  use int.ComputerDivision

  let predicate divides (d:int) (n:int)
    ensures { result <-> exists q:int. n = q * d }
  = if d = 0 then n = 0 else mod n d = 0

  lemma divides_refl: forall n:int. divides n n
  lemma divides_1_n : forall n:int. divides 1 n
  lemma divides_0   : forall n:int. divides n 0

  lemma divides_left : forall a b c: int. divides a b -> divides (c*a) (c*b)
  lemma divides_right: forall a b c: int. divides a b -> divides (a*c) (b*c)

  lemma divides_oppr: forall a b: int. divides a b -> divides a (-b)
  lemma divides_oppl: forall a b: int. divides a b -> divides (-a) b
  lemma divides_oppr_rev: forall a b: int. divides (-a) b -> divides a b
  lemma divides_oppl_rev: forall a b: int. divides a (-b) -> divides a b

  lemma divides_plusr:
    forall a b c: int. divides a b -> divides a c -> divides a (b + c)
  lemma divides_minusr:
    forall a b c: int. divides a b -> divides a c -> divides a (b - c)
  lemma divides_multl:
    forall a b c: int. divides a b -> divides a (c * b)
  lemma divides_multr:
    forall a b c: int. divides a b -> divides a (b * c)

  lemma divides_factorl: forall a b: int. divides a (b * a)
  lemma divides_factorr: forall a b: int. divides a (a * b)

  lemma divides_n_1: forall n: int. divides n 1 -> n = 1 \/ n = -1

  lemma divides_antisym:
    forall a b: int. divides a b -> divides b a -> a = b \/ a = -b

  lemma divides_trans:
    forall a b c: int. divides a b -> divides b c -> divides a c

  use int.Abs

  lemma divides_bounds:
    forall a b: int. divides a b -> b <> 0 -> abs a <= abs b

  use int.EuclideanDivision as ED

  lemma mod_divides_euclidean:
    forall a b: int. b <> 0 -> ED.mod a b = 0 -> divides b a
  lemma divides_mod_euclidean:
    forall a b: int. b <> 0 -> divides b a -> ED.mod a b = 0

  use int.ComputerDivision as CD

  lemma mod_divides_computer:
    forall a b: int. b <> 0 -> CD.mod a b = 0 -> divides b a
  lemma divides_mod_computer:
    forall a b: int. b <> 0 -> divides b a -> CD.mod a b = 0

  use Parity

  lemma even_divides: forall a: int. even a <-> divides 2 a
  lemma odd_divides: forall a: int. odd a <-> not (divides 2 a)

end

Greateast Common Divisor

module Gcd

  use export int.Int
  use Divisibility

  function gcd int int : int

  axiom gcd_nonneg: forall a b: int. 0 <= gcd a b
  axiom gcd_def1  : forall a b: int. divides (gcd a b) a
  axiom gcd_def2  : forall a b: int. divides (gcd a b) b
  axiom gcd_def3  :
    forall a b x: int. divides x a -> divides x b -> divides x (gcd a b)
  axiom gcd_unique:
    forall a b d: int.
    0 <= d -> divides d a -> divides d b ->
    (forall x: int. divides x a -> divides x b -> divides x d) ->
    d = gcd a b

  (* gcd is associative commutative *)

  clone algebra.AC with type t = int, function op = gcd

  lemma gcd_0_pos: forall a: int. 0 <= a -> gcd a 0 = a
  lemma gcd_0_neg: forall a: int. a <  0 -> gcd a 0 = -a

  lemma gcd_opp: forall a b: int. gcd a b = gcd (-a) b

  lemma gcd_euclid: forall a b q: int. gcd a b = gcd a (b - q * a)

  use int.ComputerDivision as CD

  lemma Gcd_computer_mod:
    forall a b: int [gcd b (CD.mod a b)].
    b <> 0 -> gcd b (CD.mod a b) = gcd a b

  use int.EuclideanDivision as ED

  lemma Gcd_euclidean_mod:
    forall a b: int [gcd b (ED.mod a b)].
    b <> 0 -> gcd b (ED.mod a b) = gcd a b

  lemma gcd_mult: forall a b c: int. 0 <= c -> gcd (c * a) (c * b) = c * gcd a b

end

Prime numbers

module Prime

  use export int.Int
  use Divisibility

  predicate prime (p: int) =
    2 <= p /\ forall n: int. 1 < n < p -> not (divides n p)

  lemma not_prime_1: not (prime 1)
  lemma prime_2    : prime 2
  lemma prime_3    : prime 3

  lemma prime_divisors:
    forall p: int. prime p ->
    forall d: int. divides d p -> d = 1 \/ d = -1 \/ d = p \/ d = -p

  lemma small_divisors:
    forall p: int. 2 <= p ->
    (forall d: int. 2 <= d -> prime d -> 1 < d*d <= p -> not (divides d p)) ->
    prime p

  use Parity

  lemma even_prime: forall p: int. prime p -> even p -> p = 2

  lemma odd_prime: forall p: int. prime p -> p >= 3 -> odd p

end

Coprime numbers

module Coprime

  use export int.Int
  use Divisibility
  use Gcd

  predicate coprime (a b: int) = gcd a b = 1

  use Prime

  lemma prime_coprime:
    forall p: int.
    prime p <-> 2 <= p && forall n:int. 1 <= n < p -> coprime n p

  lemma Gauss:
    forall a b c:int. divides a (b*c) /\ coprime a b -> divides a c

  lemma Euclid:
    forall p a b:int.
      prime p /\ divides p (a*b) -> divides p a \/ divides p b

  lemma gcd_coprime:
    forall a b c. coprime a b -> gcd a (b*c) = gcd a c

end