(* Title: IntDef.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
The integers as equivalence classes over nat*nat.
*)
IntDef = Equiv + NatArith +
constdefs
intrel :: "((nat * nat) * (nat * nat)) set"
"intrel == {p. EX x1 y1 x2 y2. p=((x1::nat,y1),(x2,y2)) & x1+y2 = x2+y1}"
typedef (Integ)
int = "UNIV//intrel" (quotient_def)
instance
int :: {ord, zero, one, plus, times, minus}
defs
zminus_def
"- Z == Abs_Integ(UN (x,y):Rep_Integ(Z). intrel``{(y,x)})"
constdefs
int :: nat => int
"int m == Abs_Integ(intrel `` {(m,0)})"
neg :: int => bool
"neg(Z) == EX x y. x<y & (x,y::nat):Rep_Integ(Z)"
(*For simplifying equalities*)
iszero :: int => bool
"iszero z == z = (0::int)"
defs (*of overloaded constants*)
Zero_int_def "0 == int 0"
One_int_def "1 == int 1"
zadd_def
"z + w ==
Abs_Integ(UN (x1,y1):Rep_Integ(z). UN (x2,y2):Rep_Integ(w).
intrel``{(x1+x2, y1+y2)})"
zdiff_def "z - (w::int) == z + (-w)"
zless_def "z<w == neg(z - w)"
zle_def "z <= (w::int) == ~(w < z)"
zmult_def
"z * w ==
Abs_Integ(UN (x1,y1):Rep_Integ(z). UN (x2,y2):Rep_Integ(w).
intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
end