src/ZF/Order.thy
 author clasohm Tue, 06 Feb 1996 12:27:17 +0100 changeset 1478 2b8c2a7547ab parent 1401 0c439768f45c child 1851 7b1e1c298e50 permissions -rw-r--r--
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```
(*  Title:      ZF/Order.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Orders in Zermelo-Fraenkel Set Theory
*)

Order = WF + Perm +
consts
part_ord        :: [i,i]=>o           (*Strict partial ordering*)
linear, tot_ord :: [i,i]=>o           (*Strict total ordering*)
well_ord        :: [i,i]=>o           (*Well-ordering*)
mono_map        :: [i,i,i,i]=>i       (*Order-preserving maps*)
ord_iso         :: [i,i,i,i]=>i       (*Order isomorphisms*)
pred            :: [i,i,i]=>i (*Set of predecessors*)
ord_iso_map     :: [i,i,i,i]=>i       (*Construction for linearity theorem*)

defs
part_ord_def "part_ord(A,r) == irrefl(A,r) & trans[A](r)"

linear_def   "linear(A,r) == (ALL x:A. ALL y:A. <x,y>:r | x=y | <y,x>:r)"

tot_ord_def  "tot_ord(A,r) == part_ord(A,r) & linear(A,r)"

well_ord_def "well_ord(A,r) == tot_ord(A,r) & wf[A](r)"

mono_map_def "mono_map(A,r,B,s) ==
{f: A->B. ALL x:A. ALL y:A. <x,y>:r --> <f`x,f`y>:s}"

ord_iso_def  "ord_iso(A,r,B,s) ==
{f: bij(A,B). ALL x:A. ALL y:A. <x,y>:r <-> <f`x,f`y>:s}"

pred_def     "pred(A,x,r) == {y:A. <y,x>:r}"

ord_iso_map_def
"ord_iso_map(A,r,B,s) ==
UN x:A. UN y:B. UN f: ord_iso(pred(A,x,r), r, pred(B,y,s), s).
{<x,y>}"

end
```