(* Title: HOL/UNITY/PriorityAux
ID: $Id$
Author: Sidi O Ehmety, Cambridge University Computer Laboratory
Copyright 2001 University of Cambridge
Auxiliary definitions needed in Priority.thy
*)
PriorityAux = UNITY_Main +
types vertex
arities vertex :: type
constdefs
(* symmetric closure: removes the orientation of a relation *)
symcl :: "(vertex*vertex)set=>(vertex*vertex)set"
"symcl r == r Un (r^-1)"
(* Neighbors of a vertex i *)
neighbors :: "[vertex, (vertex*vertex)set]=>vertex set"
"neighbors i r == ((r Un r^-1)``{i}) - {i}"
R :: "[vertex, (vertex*vertex)set]=>vertex set"
"R i r == r``{i}"
A :: "[vertex, (vertex*vertex)set]=>vertex set"
"A i r == (r^-1)``{i}"
(* reachable and above vertices: the original notation was R* and A* *)
reach :: "[vertex, (vertex*vertex)set]=> vertex set"
"reach i r == (r^+)``{i}"
above :: "[vertex, (vertex*vertex)set]=> vertex set"
"above i r == ((r^-1)^+)``{i}"
reverse :: "[vertex, (vertex*vertex) set]=>(vertex*vertex)set"
"reverse i r == (r - {(x,y). x=i | y=i} Int r) Un ({(x,y). x=i|y=i} Int r)^-1"
(* The original definition *)
derive1 :: "[vertex, (vertex*vertex)set, (vertex*vertex)set]=>bool"
"derive1 i r q == symcl r = symcl q &
(ALL k k'. k~=i & k'~=i -->((k,k'):r) = ((k,k'):q)) &
A i r = {} & R i q = {}"
(* Our alternative definition *)
derive :: "[vertex, (vertex*vertex)set, (vertex*vertex)set]=>bool"
"derive i r q == A i r = {} & (q = reverse i r)"
rules
(* we assume that the universe of vertices is finite *)
finite_vertex_univ "finite (UNIV :: vertex set)"
end