2 (** {1 Number theory} *)
5 (** {2 Parity properties} *)
11 predicate even (n: int) = exists k: int. n = 2 * k
12 predicate odd (n: int) = exists k: int. n = 2 * k + 1
14 lemma even_or_odd: forall n: int. even n \/ odd n
16 lemma even_not_odd: forall n: int. even n -> not (odd n)
17 lemma odd_not_even: forall n: int. odd n -> not (even n)
19 lemma even_odd: forall n: int. even n -> odd (n + 1)
20 lemma odd_even: forall n: int. odd n -> even (n + 1)
22 lemma even_even: forall n: int. even n -> even (n + 2)
23 lemma odd_odd: forall n: int. odd n -> odd (n + 2)
25 lemma even_2k: forall k: int. even (2 * k)
26 lemma odd_2k1: forall k: int. odd (2 * k + 1)
28 use int.ComputerDivision
31 forall n:int. even n <-> mod n 2 = 0
35 (** {2 Divisibility} *)
40 use int.ComputerDivision
42 let predicate divides (d:int) (n:int)
43 ensures { result <-> exists q:int. n = q * d }
44 = if d = 0 then n = 0 else mod n d = 0
46 lemma divides_refl: forall n:int. divides n n
47 lemma divides_1_n : forall n:int. divides 1 n
48 lemma divides_0 : forall n:int. divides n 0
50 lemma divides_left : forall a b c: int. divides a b -> divides (c*a) (c*b)
51 lemma divides_right: forall a b c: int. divides a b -> divides (a*c) (b*c)
53 lemma divides_oppr: forall a b: int. divides a b -> divides a (-b)
54 lemma divides_oppl: forall a b: int. divides a b -> divides (-a) b
55 lemma divides_oppr_rev: forall a b: int. divides (-a) b -> divides a b
56 lemma divides_oppl_rev: forall a b: int. divides a (-b) -> divides a b
59 forall a b c: int. divides a b -> divides a c -> divides a (b + c)
61 forall a b c: int. divides a b -> divides a c -> divides a (b - c)
63 forall a b c: int. divides a b -> divides a (c * b)
65 forall a b c: int. divides a b -> divides a (b * c)
67 lemma divides_factorl: forall a b: int. divides a (b * a)
68 lemma divides_factorr: forall a b: int. divides a (a * b)
70 lemma divides_n_1: forall n: int. divides n 1 -> n = 1 \/ n = -1
72 lemma divides_antisym:
73 forall a b: int. divides a b -> divides b a -> a = b \/ a = -b
76 forall a b c: int. divides a b -> divides b c -> divides a c
81 forall a b: int. divides a b -> b <> 0 -> abs a <= abs b
83 use int.EuclideanDivision as ED
85 lemma mod_divides_euclidean:
86 forall a b: int. b <> 0 -> ED.mod a b = 0 -> divides b a
87 lemma divides_mod_euclidean:
88 forall a b: int. b <> 0 -> divides b a -> ED.mod a b = 0
90 use int.ComputerDivision as CD
92 lemma mod_divides_computer:
93 forall a b: int. b <> 0 -> CD.mod a b = 0 -> divides b a
94 lemma divides_mod_computer:
95 forall a b: int. b <> 0 -> divides b a -> CD.mod a b = 0
99 lemma even_divides: forall a: int. even a <-> divides 2 a
100 lemma odd_divides: forall a: int. odd a <-> not (divides 2 a)
104 (** {2 Greateast Common Divisor} *)
111 function gcd int int : int
113 axiom gcd_nonneg: forall a b: int. 0 <= gcd a b
114 axiom gcd_def1 : forall a b: int. divides (gcd a b) a
115 axiom gcd_def2 : forall a b: int. divides (gcd a b) b
117 forall a b x: int. divides x a -> divides x b -> divides x (gcd a b)
120 0 <= d -> divides d a -> divides d b ->
121 (forall x: int. divides x a -> divides x b -> divides x d) ->
124 (* gcd is associative commutative *)
126 clone algebra.AC with type t = int, function op = gcd
128 lemma gcd_0_pos: forall a: int. 0 <= a -> gcd a 0 = a
129 lemma gcd_0_neg: forall a: int. a < 0 -> gcd a 0 = -a
131 lemma gcd_opp: forall a b: int. gcd a b = gcd (-a) b
133 lemma gcd_euclid: forall a b q: int. gcd a b = gcd a (b - q * a)
135 use int.ComputerDivision as CD
137 lemma Gcd_computer_mod:
138 forall a b: int [gcd b (CD.mod a b)].
139 b <> 0 -> gcd b (CD.mod a b) = gcd a b
141 use int.EuclideanDivision as ED
143 lemma Gcd_euclidean_mod:
144 forall a b: int [gcd b (ED.mod a b)].
145 b <> 0 -> gcd b (ED.mod a b) = gcd a b
147 lemma gcd_mult: forall a b c: int. 0 <= c -> gcd (c * a) (c * b) = c * gcd a b
151 (** {2 Prime numbers} *)
158 predicate prime (p: int) =
159 2 <= p /\ forall n: int. 1 < n < p -> not (divides n p)
161 lemma not_prime_1: not (prime 1)
162 lemma prime_2 : prime 2
163 lemma prime_3 : prime 3
165 lemma prime_divisors:
166 forall p: int. prime p ->
167 forall d: int. divides d p -> d = 1 \/ d = -1 \/ d = p \/ d = -p
169 lemma small_divisors:
170 forall p: int. 2 <= p ->
171 (forall d: int. 2 <= d -> prime d -> 1 < d*d <= p -> not (divides d p)) ->
176 lemma even_prime: forall p: int. prime p -> even p -> p = 2
178 lemma odd_prime: forall p: int. prime p -> p >= 3 -> odd p
182 (** {2 Coprime numbers} *)
190 predicate coprime (a b: int) = gcd a b = 1
196 prime p <-> 2 <= p && forall n:int. 1 <= n < p -> coprime n p
199 forall a b c:int. divides a (b*c) /\ coprime a b -> divides a c
203 prime p /\ divides p (a*b) -> divides p a \/ divides p b
206 forall a b c. coprime a b -> gcd a (b*c) = gcd a c