Thm* x: , n: . ((x mod n) mod n) = (x mod n) | [mod_mod] |
Thm* n: . (n mod n) = 0 | [mod_self] |
Thm* x,y: , n: . ((x+y) mod n) = (((x mod n)+(y mod n)) mod n) | [mod_add] |
Thm* x: , n: . ((-x) mod n) = if (x mod n)= 0 0 else n-(x mod n) fi | [mod_minus] |
Thm* n: . (0 mod n) = 0 | [zero_mod] |
Thm* x,y: , n: . ((x y) mod n) = (((x mod n) (y mod n)) mod n) | [mod_mul] |
Thm* a: , n: , q: , r: . a = q n+r  r < n  q = (a  n) & r = (a mod n) | [div_floor_mod_unique] |
Thm* a: , n: . a = (a  n) n+(a mod n) & (a mod n) < n | [div_floor_mod_properties] |