more renaming of theorems from _nat to _int (corresponding to a function that
was similarly renamed some time ago
Also new theorem zmult_int
(* 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 + 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}
defs
zminus_def
"- Z == Abs_Integ(UN p:Rep_Integ(Z). split (%x y. intrel^^{(y,x)}) p)"
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 = int 0"
defs
zadd_def
"z + w ==
Abs_Integ(UN p1:Rep_Integ(z). UN p2:Rep_Integ(w).
split (%x1 y1. split (%x2 y2. intrel^^{(x1+x2, y1+y2)}) p2) p1)"
zdiff_def "z - w == z + -(w::int)"
zless_def "z<w == neg(z - w)"
zle_def "z <= (w::int) == ~(w < z)"
zmult_def
"z * w ==
Abs_Integ(UN p1:Rep_Integ(z). UN p2:Rep_Integ(w). split (%x1 y1.
split (%x2 y2. intrel^^{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)"
end