Thm* r:![](FONT/nat.png) , s:{s':![](FONT/nat.png) | CoPrime(r,s') }, a,b: .
Thm* ![](FONT/down.png) x: . ((x = a mod r) & (x = b mod s)) | [chrem_exists_a] |
Thm* r,s:![](FONT/nat.png) . CoPrime(r,s) ![](FONT/eq.png) ( a,b: . x: . (x = a mod r) & (x = b mod s)) | [chrem_exists] |
Thm* r:![](FONT/nat.png) , s:{s':![](FONT/nat.png) | CoPrime(r,s') }. ![](FONT/down.png) x: . ((x = 1 mod r) & (x = 0 mod s)) | [chrem_exists_aux_a] |
Thm* r,s:![](FONT/nat.png) . CoPrime(r,s) ![](FONT/eq.png) ( x: . (x = 1 mod r) & (x = 0 mod s)) | [chrem_exists_aux] |
Thm* p: . prime(p) ![](FONT/eq.png) p = 0 & (p ~ 1) & ( a: . a | p ![](FONT/eq.png) (a ~ 1) (a ~ p)) | [prime_elim] |
Thm* a: . atomic(a) ![](FONT/if_big.png) (a ~ 1) & ( b: . b | a ![](FONT/eq.png) (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* ![](FONT/eq.png)
Thm* GCD(x;c;y)
Thm* ![](FONT/eq.png)
Thm* y | a & y | b & y | c & ( z: . z | a ![](FONT/eq.png) z | b ![](FONT/eq.png) z | c ![](FONT/eq.png) z | y) | [gcd_of_triple] |
Thm* a: , b:![](FONT/nat.png) . a | b & b | a ![](FONT/if_big.png) a<b & a | b | [pdivisor_bound] |
Thm* a,b: . a | b & b | a ![](FONT/if_big.png) a = b | [assoc_reln] |
Def prime(a) == a = 0 & (a ~ 1) & ( b,c: . a | b c ![](FONT/eq.png) a | b a | c) | [prime] |
Def atomic(a) == a = 0 & (a ~ 1) & reducible(a) | [atomic] |
Def reducible(a) == b,c:![](FONT/int.png) ![](FONT/minus.png) . (b ~ 1) & (c ~ 1) & a = b c | [reducible] |
Def GCD(a;b;y) == y | a & y | b & ( z: . z | a & z | b ![](FONT/eq.png) z | y) | [gcd_p] |
Def a ~ b == a | b & b | a | [assoced] |