(* Title: Integ.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
The integers as equivalence classes over nat*nat.
*)
Integ = Equiv + Arith +
constdefs
intrel :: "((nat * nat) * (nat * nat)) set"
"intrel == {p. ? x1 y1 x2 y2. p=((x1::nat,y1),(x2,y2)) & x1+y2 = x2+y1}"
typedef (Integ)
int = "{x::(nat*nat).True}/intrel" (Equiv.quotient_def)
instance
int :: {ord, plus, times, minus}
constdefs
znat :: nat => int ("$# _" [80] 80)
"$# m == Abs_Integ(intrel ^^ {(m,0)})"
zminus :: int => int ("$~ _" [80] 80)
"$~ Z == Abs_Integ(UN p:Rep_Integ(Z). split (%x y. intrel^^{(y,x)}) p)"
znegative :: int => bool
"znegative(Z) == EX x y. x<y & (x,y::nat):Rep_Integ(Z)"
zmagnitude :: int => int
"zmagnitude(Z) == Abs_Integ(UN p:Rep_Integ(Z).
split (%x y. intrel^^{((y-x) + (x-y),0)}) p)"
zpred :: int=>int
"zpred(Z) == Z - $# Suc(0)"
zsuc :: int=>int
"zsuc(Z) == Z + $# Suc(0)"
defs
zadd_def
"Z1 + Z2 ==
Abs_Integ(UN p1:Rep_Integ(Z1). UN p2:Rep_Integ(Z2).
split (%x1 y1. split (%x2 y2. intrel^^{(x1+x2, y1+y2)}) p2) p1)"
zdiff_def "Z1 - Z2 == Z1 + zminus(Z2)"
zless_def "Z1<Z2 == znegative(Z1 - Z2)"
zle_def "Z1 <= (Z2::int) == ~(Z2 < Z1)"
zmult_def
"Z1 * Z2 ==
Abs_Integ(UN p1:Rep_Integ(Z1). UN p2:Rep_Integ(Z2). split (%x1 y1.
split (%x2 y2. intrel^^{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)"
end