--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/IOA/IOA.thy Wed Apr 30 11:56:17 1997 +0200
@@ -0,0 +1,196 @@
+(* Title: HOL/IOA/meta_theory/IOA.thy
+ ID: $Id$
+ Author: Tobias Nipkow & Konrad Slind
+ Copyright 1994 TU Muenchen
+
+The I/O automata of Lynch and Tuttle.
+*)
+
+IOA = Asig + Option +
+
+types
+ 'a seq = "nat => 'a"
+ 'a oseq = "nat => 'a option"
+ ('a,'b)execution = "'a oseq * 'b seq"
+ ('a,'s)transition = "('s * 'a * 's)"
+ ('a,'s)ioa = "'a signature * 's set * ('a,'s)transition set"
+
+consts
+
+ (* IO automata *)
+ state_trans::"['action signature, ('action,'state)transition set] => bool"
+ asig_of ::"('action,'state)ioa => 'action signature"
+ starts_of ::"('action,'state)ioa => 'state set"
+ trans_of ::"('action,'state)ioa => ('action,'state)transition set"
+ IOA ::"('action,'state)ioa => bool"
+
+ (* Executions, schedules, and traces *)
+
+ is_execution_fragment,
+ has_execution ::"[('action,'state)ioa, ('action,'state)execution] => bool"
+ executions :: "('action,'state)ioa => ('action,'state)execution set"
+ mk_trace :: "[('action,'state)ioa, 'action oseq] => 'action oseq"
+ reachable :: "[('action,'state)ioa, 'state] => bool"
+ invariant :: "[('action,'state)ioa, 'state=>bool] => bool"
+ has_trace :: "[('action,'state)ioa, 'action oseq] => bool"
+ traces :: "('action,'state)ioa => 'action oseq set"
+ NF :: "'a oseq => 'a oseq"
+
+ (* Composition of action signatures and automata *)
+ compatible_asigs ::"('a => 'action signature) => bool"
+ asig_composition ::"('a => 'action signature) => 'action signature"
+ compatible_ioas ::"('a => ('action,'state)ioa) => bool"
+ ioa_composition ::"('a => ('action, 'state)ioa) =>('action,'a => 'state)ioa"
+
+ (* binary composition of action signatures and automata *)
+ compat_asigs ::"['action signature, 'action signature] => bool"
+ asig_comp ::"['action signature, 'action signature] => 'action signature"
+ compat_ioas ::"[('action,'s)ioa, ('action,'t)ioa] => bool"
+ "||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr 10)
+
+ (* Filtering and hiding *)
+ filter_oseq :: "('a => bool) => 'a oseq => 'a oseq"
+
+ restrict_asig :: "['a signature, 'a set] => 'a signature"
+ restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
+
+ (* Notions of correctness *)
+ ioa_implements :: "[('action,'state1)ioa, ('action,'state2)ioa] => bool"
+
+ (* Instantiation of abstract IOA by concrete actions *)
+ rename:: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
+
+defs
+
+state_trans_def
+ "state_trans asig R ==
+ (!triple. triple:R --> fst(snd(triple)):actions(asig)) &
+ (!a. (a:inputs(asig)) --> (!s1. ? s2. (s1,a,s2):R))"
+
+
+asig_of_def "asig_of == fst"
+starts_of_def "starts_of == (fst o snd)"
+trans_of_def "trans_of == (snd o snd)"
+
+ioa_def
+ "IOA(ioa) == (is_asig(asig_of(ioa)) &
+ (~ starts_of(ioa) = {}) &
+ state_trans (asig_of ioa) (trans_of ioa))"
+
+
+(* An execution fragment is modelled with a pair of sequences:
+ * the first is the action options, the second the state sequence.
+ * Finite executions have None actions from some point on.
+ *******)
+is_execution_fragment_def
+ "is_execution_fragment A ex ==
+ let act = fst(ex); state = snd(ex)
+ in !n a. (act(n)=None --> state(Suc(n)) = state(n)) &
+ (act(n)=Some(a) --> (state(n),a,state(Suc(n))):trans_of(A))"
+
+
+executions_def
+ "executions(ioa) == {e. snd e 0:starts_of(ioa) &
+ is_execution_fragment ioa e}"
+
+
+reachable_def
+ "reachable ioa s == (? ex:executions(ioa). ? n. (snd ex n) = s)"
+
+
+invariant_def "invariant A P == (!s. reachable A s --> P(s))"
+
+
+(* Restrict the trace to those members of the set s *)
+filter_oseq_def
+ "filter_oseq p s ==
+ (%i.case s(i)
+ of None => None
+ | Some(x) => if p x then Some x else None)"
+
+
+mk_trace_def
+ "mk_trace(ioa) == filter_oseq(%a.a:externals(asig_of(ioa)))"
+
+
+(* Does an ioa have an execution with the given trace *)
+has_trace_def
+ "has_trace ioa b ==
+ (? ex:executions(ioa). b = mk_trace ioa (fst ex))"
+
+normal_form_def
+ "NF(tr) == @nf. ? f. mono(f) & (!i. nf(i)=tr(f(i))) &
+ (!j. j ~: range(f) --> nf(j)= None) &
+ (!i. nf(i)=None --> (nf (Suc i)) = None) "
+
+(* All the traces of an ioa *)
+
+ traces_def
+ "traces(ioa) == {trace. ? tr. trace=NF(tr) & has_trace ioa tr}"
+
+(*
+ traces_def
+ "traces(ioa) == {tr. has_trace ioa tr}"
+*)
+
+compat_asigs_def
+ "compat_asigs a1 a2 ==
+ (((outputs(a1) Int outputs(a2)) = {}) &
+ ((internals(a1) Int actions(a2)) = {}) &
+ ((internals(a2) Int actions(a1)) = {}))"
+
+
+compat_ioas_def
+ "compat_ioas ioa1 ioa2 == compat_asigs (asig_of(ioa1)) (asig_of(ioa2))"
+
+
+asig_comp_def
+ "asig_comp a1 a2 ==
+ (((inputs(a1) Un inputs(a2)) - (outputs(a1) Un outputs(a2)),
+ (outputs(a1) Un outputs(a2)),
+ (internals(a1) Un internals(a2))))"
+
+
+par_def
+ "(ioa1 || ioa2) ==
+ (asig_comp (asig_of ioa1) (asig_of ioa2),
+ {pr. fst(pr):starts_of(ioa1) & snd(pr):starts_of(ioa2)},
+ {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
+ in (a:actions(asig_of(ioa1)) | a:actions(asig_of(ioa2))) &
+ (if a:actions(asig_of(ioa1)) then
+ (fst(s),a,fst(t)):trans_of(ioa1)
+ else fst(t) = fst(s))
+ &
+ (if a:actions(asig_of(ioa2)) then
+ (snd(s),a,snd(t)):trans_of(ioa2)
+ else snd(t) = snd(s))})"
+
+
+restrict_asig_def
+ "restrict_asig asig actns ==
+ (inputs(asig) Int actns, outputs(asig) Int actns,
+ internals(asig) Un (externals(asig) - actns))"
+
+
+restrict_def
+ "restrict ioa actns ==
+ (restrict_asig (asig_of ioa) actns, starts_of(ioa), trans_of(ioa))"
+
+
+ioa_implements_def
+ "ioa_implements ioa1 ioa2 ==
+ ((inputs(asig_of(ioa1)) = inputs(asig_of(ioa2))) &
+ (outputs(asig_of(ioa1)) = outputs(asig_of(ioa2))) &
+ traces(ioa1) <= traces(ioa2))"
+
+rename_def
+"rename ioa ren ==
+ (({b. ? x. Some(x)= ren(b) & x : inputs(asig_of(ioa))},
+ {b. ? x. Some(x)= ren(b) & x : outputs(asig_of(ioa))},
+ {b. ? x. Some(x)= ren(b) & x : internals(asig_of(ioa))}),
+ starts_of(ioa) ,
+ {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
+ in
+ ? x. Some(x) = ren(a) & (s,x,t):trans_of(ioa)})"
+
+end