(* Title: Integ.thy
ID: $Id$
Authors: Riccardo Mattolini, Dip. Sistemi e Informatica
Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 Universita' di Firenze
Copyright 1993 University of Cambridge
The integers as equivalence classes over nat*nat.
*)
Integ = Equiv + Arith +
consts
intrel :: "((nat * nat) * (nat * nat)) set"
defs
intrel_def
"intrel == {p. ? x1 y1 x2 y2. p=<<x1::nat,y1>,<x2,y2>> & x1+y2 = x2+y1}"
subtype (Integ)
int = "{x::(nat*nat).True}/intrel" ("quotient_def")
arities int :: ord
int :: plus
int :: times
int :: minus
consts
zNat :: "nat set"
znat :: "nat => int" ("$# _" [80] 80)
zminus :: "int => int" ("$~ _" [80] 80)
znegative :: "int => bool"
zmagnitude :: "int => int"
zdiv,zmod :: "[int,int]=>int" (infixl 70)
zpred,zsuc :: "int=>int"
defs
zNat_def "zNat == {x::nat. True}"
znat_def "$# m == Abs_Integ(intrel ^^ {<m,0>})"
zminus_def
"$~ Z == Abs_Integ(UN p:Rep_Integ(Z). split (%x y. intrel^^{<y,x>}) p)"
znegative_def
"znegative(Z) == EX x y. x<y & <x,y::nat>:Rep_Integ(Z)"
zmagnitude_def
"zmagnitude(Z) == Abs_Integ(UN p:Rep_Integ(Z).split (%x y. intrel^^{<(y-x) + (x-y),0>}) p)"
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)"
zdiv_def
"Z1 zdiv Z2 == \
\ Abs_Integ(UN p1:Rep_Integ(Z1). UN p2:Rep_Integ(Z2). split (%x1 y1. \
\ split (%x2 y2. intrel^^{<(x1-y1)div(x2-y2)+(y1-x1)div(y2-x2), \
\ (x1-y1)div(y2-x2)+(y1-x1)div(x2-y2)>}) p2) p1)"
zmod_def
"Z1 zmod Z2 == \
\ Abs_Integ(UN p1:Rep_Integ(Z1).UN p2:Rep_Integ(Z2).split (%x1 y1. \
\ split (%x2 y2. intrel^^{<(x1-y1)mod((x2-y2)+(y2-x2)), \
\ (x1-y1)mod((x2-y2)+(x2-y2))>}) p2) p1)"
zsuc_def "zsuc(Z) == Z + $# Suc(0)"
zpred_def "zpred(Z) == Z - $# Suc(0)"
end