Thm* p: . prime(p)  ( a1,a2: . p | a1 a2  p | a1 p | a2) | [prime_divs_prod] |
Thm* a1,a2,b: . CoPrime(a1,a2)  a1 | b  a2 | b  a1 a2 | b | [coprime_divisors_prod] |
Thm* a,p: . prime(p)  (CoPrime(p,a)  p | a) | [coprime_iff_ndivides] |
Thm* a,b: . CoPrime(a,b)  ( c: . c | a  c | b  (c ~ 1)) | [coprime_elim] |
Thm* a,b: . ( c: . c | a  c | b  c | 1)  CoPrime(a,b) | [coprime_intro] |
Thm* p: . prime(p)  p = 0 & (p ~ 1) & ( a: . a | p  (a ~ 1) (a ~ p)) | [prime_elim] |
Thm* a:  , b: . a b | a  (b ~ 1) | [self_divisor_mul] |
Thm* a: . atomic(a)  (a ~ 1) & ( b: . b | a  (b ~ 1) (b ~ a)) | [atomic_char] |
Thm* a,b,c: . c | a  c | b  c | gcd(a;b) | [gcd_is_gcd] |
Thm* a,b: . gcd(a;b) | b | [gcd_is_divisor_2] |
Thm* a,b: . gcd(a;b) | a | [gcd_is_divisor_1] |
Thm* a,b,c,x,y: .
Thm* GCD(a;b;x)
Thm* 
Thm* GCD(x;c;y)
Thm* 
Thm* y | a & y | b & y | c & ( z: . z | a  z | b  z | c  z | y) | [gcd_of_triple] |
Thm* a,b: . a | b  ( c: . b = a c  ) | [divides_nchar] |
Thm* a: , b: . a | b & b | a  a<b & a | b | [pdivisor_bound] |
Thm* a: . a | 1  (a ~ 1) | [unit_chars] |
Thm* a,a',b,b': . (a ~ a')  (b ~ b')  (a | b  a' | b') | [divides_functionality_wrt_assoced] |
Thm* 3 | 6 | [divides_instance] |
Thm* a,b: . Dec(a | b) | [decidable__divides] |
Thm* a: , n:  . n | a  (a n) n = a | [divides_iff_div_exact] |
Thm* a: , b:  . b | a  (a rem b) = 0 | [divides_iff_rem_zero] |
Thm* a: , b: . a | b  a b | [divisor_bound] |
Thm* a,b: . a | b  ( n: . n a | n b) | [divides_mul] |
Thm* a,b,c: . a | b  a | b c | [divisor_of_mul] |
Thm* a,b: . a | b  a | -b | [divisor_of_minus] |
Thm* a,b1,b2: . a | b1  a | b2  a | b1+b2 | [divisor_of_sum] |
Thm* a,b: . a | b & b | a  a = b | [assoc_reln] |
Thm* a,b: . a | b  b | a  a = b | [divides_anti_sym] |
Thm* a,b: . a | b  b | a  a = b | [divides_anti_sym_n] |
Thm* Preorder( ;x,y.x | y) | [divides_preorder] |
Thm* a,b,c: . a | b  b | c  a | c | [divides_transitivity] |
Thm* a: . a | a | [divides_reflexivity] |
Thm* a,b: . |a| | |b|  a | b | [divides_of_absvals] |
Thm* b: . b | 1  b = 1 | [only_pm_one_divs_one] |
Thm* a: , b: . a | b  a b | [divisors_bound] |
Thm* a,b: . a | b  a | -b | [divides_invar_2] |
Thm* a,b: . a | b  -a | b | [divides_invar_1] |
Thm* b: . b | 0 | [any_divs_zero] |
Thm* a: . 1 | a | [one_divs_any] |
Thm* a: . 0 | a  a = 0 | [zero_divs_only_zero] |
Def a = b mod m == m | a-b | [eqmod] |
Def prime(a) == a = 0 & (a ~ 1) & ( b,c: . a | b c  a | b a | c) | [prime] |
Def GCD(a;b;y) == y | a & y | b & ( z: . z | a & z | b  z | y) | [gcd_p] |
Def a ~ b == a | b & b | a | [assoced] |