(* 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 + "mono" +
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