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More Arithmetic

|x| := if 0 <= x then x else -x;

sign(x) := if x = 0 then 0 else if 0 <= x then 1 else -1;

x div y := such q: exists r:

|r| < |y| /\  x=q*y+r /\  (sign(r) = 0 \/ sign(r)=sign(y));

x mod y := such r: exists q:

|r| < |y| /\  x=q*y+r /\  (sign(r) = 0 \/ sign(r)=sign(y)).