src/ZF/Perm.thy
author clasohm
Tue Feb 06 12:27:17 1996 +0100 (1996-02-06)
changeset 1478 2b8c2a7547ab
parent 1401 0c439768f45c
child 1806 12708740f58d
permissions -rw-r--r--
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     1 (*  Title:      ZF/perm
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 
     6 The theory underlying permutation groups
     7   -- Composition of relations, the identity relation
     8   -- Injections, surjections, bijections
     9   -- Lemmas for the Schroeder-Bernstein Theorem
    10 *)
    11 
    12 Perm = ZF + "mono" +
    13 consts
    14     O           ::      [i,i]=>i      (infixr 60)
    15     id          ::      i=>i
    16     inj,surj,bij::      [i,i]=>i
    17 
    18 defs
    19 
    20     (*composition of relations and functions; NOT Suppes's relative product*)
    21     comp_def    "r O s == {xz : domain(s)*range(r) . 
    22                                 EX x y z. xz=<x,z> & <x,y>:s & <y,z>:r}"
    23 
    24     (*the identity function for A*)
    25     id_def      "id(A) == (lam x:A. x)"
    26 
    27     (*one-to-one functions from A to B*)
    28     inj_def      "inj(A,B) == { f: A->B. ALL w:A. ALL x:A. f`w=f`x --> w=x}"
    29 
    30     (*onto functions from A to B*)
    31     surj_def    "surj(A,B) == { f: A->B . ALL y:B. EX x:A. f`x=y}"
    32 
    33     (*one-to-one and onto functions*)
    34     bij_def     "bij(A,B) == inj(A,B) Int surj(A,B)"
    35 
    36 end