src/ZF/Perm.thy

Installation of new simplfier. Previously appeared to set up the old
simplifier to rewrite with the partial ordering [=, something not possible
with the new simplifier. But such rewriting appears not to have actually
been used, and there were few complications. In terms.ML setloop was used
to avoid infinite rewriting with the letrec rule. Congruence rules were
deleted, and an occasional SIMP_TAC had to become asm_simp_tac.

(*  Title: 	ZF/perm
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge

The theory underlying permutation groups
  -- Composition of relations, the identity relation
  -- Injections, surjections, bijections
  -- Lemmas for the Schroeder-Bernstein Theorem
*)

Perm = ZF +
consts
    O    	::      "[i,i]=>i"      (infixr 60)
    id  	::      "i=>i"
    inj,surj,bij::      "[i,i]=>i"

rules

    (*composition of relations and functions; NOT Suppes's relative product*)
    comp_def	"r O s == {xz : domain(s)*range(r) . \
\                  		EX x y z. xz=<x,z> & <x,y>:s & <y,z>:r}"

    (*the identity function for A*)
    id_def	"id(A) == (lam x:A. x)"

    (*one-to-one functions from A to B*)
    inj_def      "inj(A,B) == { f: A->B. ALL w:A. ALL x:A. f`w=f`x --> w=x}"

    (*onto functions from A to B*)
    surj_def	"surj(A,B) == { f: A->B . ALL y:B. EX x:A. f`x=y}"

    (*one-to-one and onto functions*)
    bij_def	"bij(A,B) == inj(A,B) Int surj(A,B)"

end