src/ZF/ex/Integ.thy
author lcp
Tue Aug 16 18:58:42 1994 +0200 (1994-08-16)
changeset 532 851df239ac8b
parent 29 4ec9b266ccd1
child 753 ec86863e87c8
permissions -rw-r--r--
ZF/Makefile,ROOT.ML, ZF/ex/Integ.thy: updated for EquivClass
     1 (*  Title: 	ZF/ex/integ.thy
     2     ID:         $Id$
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 The integers as equivalence classes over nat*nat.
     7 *)
     8 
     9 Integ = EquivClass + Arith +
    10 consts
    11     intrel,integ::      "i"
    12     znat	::	"i=>i"		("$# _" [80] 80)
    13     zminus	::	"i=>i"		("$~ _" [80] 80)
    14     znegative	::	"i=>o"
    15     zmagnitude	::	"i=>i"
    16     "$*"        ::      "[i,i]=>i"      (infixl 70)
    17     "$'/"       ::      "[i,i]=>i"      (infixl 70) 
    18     "$'/'/"     ::      "[i,i]=>i"      (infixl 70)
    19     "$+"	::      "[i,i]=>i"      (infixl 65)
    20     "$-"        ::      "[i,i]=>i"      (infixl 65)
    21     "$<"	:: 	"[i,i]=>o"  	(infixl 50)
    22 
    23 rules
    24 
    25     intrel_def
    26      "intrel == {p:(nat*nat)*(nat*nat). 		\
    27 \        EX x1 y1 x2 y2. p=<<x1,y1>,<x2,y2>> & x1#+y2 = x2#+y1}"
    28 
    29     integ_def   "integ == (nat*nat)/intrel"
    30     
    31     znat_def	"$# m == intrel `` {<m,0>}"
    32     
    33     zminus_def	"$~ Z == UN p:Z. split(%x y. intrel``{<y,x>}, p)"
    34     
    35     znegative_def
    36 	"znegative(Z) == EX x y. x<y & y:nat & <x,y>:Z"
    37     
    38     zmagnitude_def
    39 	"zmagnitude(Z) == UN p:Z. split(%x y. (y#-x) #+ (x#-y), p)"
    40     
    41     zadd_def
    42      "Z1 $+ Z2 == \
    43 \       UN p1:Z1. UN p2:Z2. split(%x1 y1. split(%x2 y2. 		\
    44 \                                         intrel``{<x1#+x2, y1#+y2>}, p2), p1)"
    45     
    46     zdiff_def   "Z1 $- Z2 == Z1 $+ zminus(Z2)"
    47     zless_def	"Z1 $< Z2 == znegative(Z1 $- Z2)"
    48     
    49     zmult_def
    50      "Z1 $* Z2 == \
    51 \       UN p1:Z1. UN p2:Z2.  split(%x1 y1. split(%x2 y2. 	\
    52 \                   intrel``{<x1#*x2 #+ y1#*y2, x1#*y2 #+ y1#*x2>}, p2), p1)"
    53     
    54  end