Thm* r:, s:{s':| CoPrime(r,s') }, a,b:.
Thm* x:. ((x = a mod r) & (x = b mod s)) | [chrem_exists_a] |
Thm* r,s:. CoPrime(r,s) (a,b:. x:. (x = a mod r) & (x = b mod s)) | [chrem_exists] |
Thm* r:, s:{s':| CoPrime(r,s') }. x:. ((x = 1 mod r) & (x = 0 mod s)) | [chrem_exists_aux_a] |
Thm* r,s:. CoPrime(r,s) (x:. (x = 1 mod r) & (x = 0 mod s)) | [chrem_exists_aux] |
Thm* p:. prime(p) p = 0 & (p ~ 1) & (a:. a | p (a ~ 1) (a ~ p)) | [prime_elim] |
Thm* a:. atomic(a) (a ~ 1) & (b:. b | a (b ~ 1) (b ~ a)) | [atomic_char] |
Thm* a,b:. y:. GCD(a;b;y) & gcd(a;b) = y | [gcd_elim] |
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 & b | a a<b & a | b | [pdivisor_bound] |
Thm* a,b:. a | b & b | a a = b | [assoc_reln] |
Def prime(a) == a = 0 & (a ~ 1) & (b,c:. a | bc a | b a | c) | [prime] |
Def atomic(a) == a = 0 & (a ~ 1) & reducible(a) | [atomic] |
Def reducible(a) == b,c:. (b ~ 1) & (c ~ 1) & a = bc | [reducible] |
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] |