(* Title: HOL/UNITY/Reach.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Reachability in Directed Graphs. From Chandy and Misra, section 6.4.
*)
Reach = FP + Traces + SubstAx +
types vertex
state = "vertex=>bool"
arities vertex :: term
consts
init :: "vertex"
edges :: "(vertex*vertex) set"
constdefs
asgt :: "[vertex,vertex] => (state*state) set"
"asgt u v == {(s,s'). s' = s(v:= s u | s v)}"
Rprg :: state program
"Rprg == mk_program ({%v. v=init}, UN (u,v): edges. {asgt u v})"
reach_invariant :: state set
"reach_invariant == {s. (ALL v. s v --> (init, v) : edges^*) & s init}"
fixedpoint :: state set
"fixedpoint == {s. ALL (u,v): edges. s u --> s v}"
metric :: state => nat
"metric s == card {v. ~ s v}"
rules
(*We assume that the set of vertices is finite*)
finite_graph "finite (UNIV :: vertex set)"
end