src/ZF/Perm.thy

UNIV now a constant; UNION1, INTER1 now translations and no longer have
separate rules for themselves

(*  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 = upair + mono + func +
consts
  O     :: [i,i]=>i      (infixr 60)

defs
  (*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}"

constdefs
  (*the identity function for A*)
  id    :: i=>i
  "id(A) == (lam x:A. x)"

  (*one-to-one functions from A to B*)
  inj   :: [i,i]=>i
  "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  :: [i,i]=>i
  "surj(A,B) == { f: A->B . ALL y:B. EX x:A. f`x=y}"

  (*one-to-one and onto functions*)
  bij   :: [i,i]=>i
  "bij(A,B) == inj(A,B) Int surj(A,B)"

end