src/ZF/ex/integ.thy

Initial revision

(*  Title: 	ZF/ex/integ.thy
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

The integers as equivalence classes over nat*nat.
*)

Integ = Equiv + Arith +
consts
    intrel,integ::      "i"
    znat	::	"i=>i"		("$# _" [80] 80)
    zminus	::	"i=>i"		("$~ _" [80] 80)
    znegative	::	"i=>o"
    zmagnitude	::	"i=>i"
    "$*"        ::      "[i,i]=>i"      (infixl 70)
    "$'/"       ::      "[i,i]=>i"      (infixl 70) 
    "$'/'/"     ::      "[i,i]=>i"      (infixl 70)
    "$+"	::      "[i,i]=>i"      (infixl 65)
    "$-"        ::      "[i,i]=>i"      (infixl 65)
    "$<"	:: 	"[i,i]=>o"  	(infixl 50)

rules

    intrel_def
     "intrel == {p:(nat*nat)*(nat*nat). 		\
\        EX x1 y1 x2 y2. p=<<x1,y1>,<x2,y2>> & x1#+y2 = x2#+y1}"

    integ_def   "integ == (nat*nat)/intrel"
    
    znat_def	"$# m == intrel `` {<m,0>}"
    
    zminus_def	"$~ Z == UN p:Z. split(%x y. intrel``{<y,x>}, p)"
    
    znegative_def
	"znegative(Z) == EX x y. x:y & y:nat & <x,y>:Z"
    
    zmagnitude_def
	"zmagnitude(Z) == UN p:Z. split(%x y. (y#-x) #+ (x#-y), p)"
    
    zadd_def
     "Z1 $+ Z2 == \
\       UN p1:Z1. UN p2: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)"
    
    zmult_def
     "Z1 $* Z2 == \
\       UN p1:Z1. UN p2:Z2.  split(%x1 y1. split(%x2 y2. 	\
\                   intrel``{<x1#*x2 #+ y1#*y2, x1#*y2 #+ y1#*x2>}, p2), p1)"
    
 end