author clasohm
Tue, 06 Feb 1996 12:27:17 +0100
changeset 1478 2b8c2a7547ab
parent 1401 0c439768f45c
child 1806 12708740f58d
permissions -rw-r--r--
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(*  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" +
    O           ::      [i,i]=>i      (infixr 60)
    id          ::      i=>i
    inj,surj,bij::      [i,i]=>i


    (*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)"