src/ZF/arith.thy

Installation of new simplifier for ZF. Deleted all congruence rules not
involving local assumptions. NB the congruence rules for Sigma and Pi
(dependent type constructions) cause difficulties and are not used by
default.

(*  Title: 	ZF/arith.thy
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

Arithmetic operators and their definitions
*)

Arith = Epsilon +
consts
    rec  :: "[i, i, [i,i]=>i]=>i"
    "#*" :: "[i,i]=>i"      		(infixl 70)
    div  :: "[i,i]=>i"      		(infixl 70) 
    mod  :: "[i,i]=>i"      		(infixl 70)
    "#+" :: "[i,i]=>i"      		(infixl 65)
    "#-" :: "[i,i]=>i"      		(infixl 65)

rules
    rec_def  "rec(k,a,b) ==  transrec(k, %n f. nat_case(a, %m. b(m, f`m), n))"

    add_def  "m#+n == rec(m, n, %u v.succ(v))"
    diff_def "m#-n == rec(n, m, %u v. rec(v, 0, %x y.x))"
    mult_def "m#*n == rec(m, 0, %u v. n #+ v)"
    mod_def  "m mod n == transrec(m, %j f. if(j:n, j, f`(j#-n)))"
    div_def  "m div n == transrec(m, %j f. if(j:n, 0, succ(f`(j#-n))))"
end