Thm* p:. prime(p) (a1,a2:. p | a1a2 p | a1 p | a2) | [prime_divs_prod] |
Thm* a1,a2,b:. CoPrime(a1,a2) a1 | b a2 | b a1a2 | 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:. ab | 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 = ac ) | [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 ab | [divisor_bound] |
Thm* a,b:. a | b (n:. na | nb) | [divides_mul] |
Thm* a,b,c:. a | b a | bc | [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 ab | [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 | bc 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] |