(* Title: HOL/BCV/DFA_Framework.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 2000 TUM
The relationship between dataflow analysis and a welltyped-insruction predicate.
*)
DFA_Framework = Listn +
constdefs
stable :: 's ord =>
(nat => 's => 's)
=> (nat => nat list) => 's list => nat => bool
"stable r step succs ss p == !q:set(succs p). step p (ss!p) <=_r ss!q"
stables :: 's ord => (nat => 's => 's)
=> (nat => nat list) => 's list => bool
"stables r step succs ss == !p<size ss. stable r step succs ss p"
is_dfa :: 's ord
=> ('s list => 's list)
=> (nat => 's => 's)
=> (nat => nat list)
=> nat => 's set => bool
"is_dfa r dfa step succs n A == !ss : list n A.
dfa ss : list n A & stables r step succs (dfa ss) & ss <=[r] dfa ss &
(!ts: list n A. ss <=[r] ts & stables r step succs ts
--> dfa ss <=[r] ts)"
is_bcv :: 's ord => 's => ('s list => nat => bool)
=> nat => 's set => ('s list => 's list) => bool
"is_bcv r T wti n A bcv == !ss : list n A.
(!p<n. (bcv ss)!p ~= T) =
(? ts: list n A. ss <=[r] ts & welltyping T wti ts)"
wti_is_stable_topless ::
's ord => 's
=> (nat => 's => 's)
=> ('s list => nat => bool)
=> (nat => nat list)
=> nat => 's set => bool
"wti_is_stable_topless r T step wti succs n A == !ss p.
ss : list n A & (!p<n. ss!p ~= T) & p < n -->
wti ss p = stable r step succs ss p"
welltyping :: 's => ('s list => nat => bool) => 's list => bool
"welltyping T wti ts == !p<size(ts). ts!p ~= T & wti ts p"
end