clasohm@0: (*  Title: 	ZF/ex/integ.thy
clasohm@0:     ID:         $Id$
clasohm@0:     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
clasohm@0:     Copyright   1993  University of Cambridge
clasohm@0: 
clasohm@0: The integers as equivalence classes over nat*nat.
clasohm@0: *)
clasohm@0: 
lcp@532: Integ = EquivClass + Arith +
clasohm@0: consts
clasohm@0:     intrel,integ::      "i"
clasohm@0:     znat	::	"i=>i"		("$# _" [80] 80)
clasohm@0:     zminus	::	"i=>i"		("$~ _" [80] 80)
clasohm@0:     znegative	::	"i=>o"
clasohm@0:     zmagnitude	::	"i=>i"
clasohm@0:     "$*"        ::      "[i,i]=>i"      (infixl 70)
clasohm@0:     "$'/"       ::      "[i,i]=>i"      (infixl 70) 
clasohm@0:     "$'/'/"     ::      "[i,i]=>i"      (infixl 70)
clasohm@0:     "$+"	::      "[i,i]=>i"      (infixl 65)
clasohm@0:     "$-"        ::      "[i,i]=>i"      (infixl 65)
clasohm@0:     "$<"	:: 	"[i,i]=>o"  	(infixl 50)
clasohm@0: 
clasohm@0: rules
clasohm@0: 
clasohm@0:     intrel_def
clasohm@0:      "intrel == {p:(nat*nat)*(nat*nat). 		\
clasohm@0: \        EX x1 y1 x2 y2. p=<<x1,y1>,<x2,y2>> & x1#+y2 = x2#+y1}"
clasohm@0: 
clasohm@0:     integ_def   "integ == (nat*nat)/intrel"
clasohm@0:     
clasohm@0:     znat_def	"$# m == intrel `` {<m,0>}"
clasohm@0:     
clasohm@0:     zminus_def	"$~ Z == UN p:Z. split(%x y. intrel``{<y,x>}, p)"
clasohm@0:     
clasohm@0:     znegative_def
lcp@29: 	"znegative(Z) == EX x y. x<y & y:nat & <x,y>:Z"
clasohm@0:     
clasohm@0:     zmagnitude_def
clasohm@0: 	"zmagnitude(Z) == UN p:Z. split(%x y. (y#-x) #+ (x#-y), p)"
clasohm@0:     
clasohm@0:     zadd_def
clasohm@0:      "Z1 $+ Z2 == \
clasohm@0: \       UN p1:Z1. UN p2:Z2. split(%x1 y1. split(%x2 y2. 		\
clasohm@0: \                                         intrel``{<x1#+x2, y1#+y2>}, p2), p1)"
clasohm@0:     
clasohm@0:     zdiff_def   "Z1 $- Z2 == Z1 $+ zminus(Z2)"
clasohm@0:     zless_def	"Z1 $< Z2 == znegative(Z1 $- Z2)"
clasohm@0:     
clasohm@0:     zmult_def
clasohm@0:      "Z1 $* Z2 == \
clasohm@0: \       UN p1:Z1. UN p2:Z2.  split(%x1 y1. split(%x2 y2. 	\
clasohm@0: \                   intrel``{<x1#*x2 #+ y1#*y2, x1#*y2 #+ y1#*x2>}, p2), p1)"
clasohm@0:     
clasohm@0:  end