src/ZF/Perm.thy
 author paulson Fri, 16 Feb 1996 18:00:47 +0100 changeset 1512 ce37c64244c0 parent 1478 2b8c2a7547ab child 1806 12708740f58d permissions -rw-r--r--
Elimination of fully-functorial style. Type tactic changed to a type abbrevation (from a datatype). Constructor tactic and function apply deleted.
```
(*  Title:      ZF/perm
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

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

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}"

(*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
```