added a fresh_left lemma that contains all instantiation
for the various atom-types.
(* Title: HOL/IOA/IOA.thy
ID: $Id$
Author: Tobias Nipkow & Konrad Slind
Copyright 1994 TU Muenchen
*)
header {* The I/O automata of Lynch and Tuttle *}
theory IOA
imports Asig
begin
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 ::"[('action,'state)ioa, ('action,'state)execution] => bool"
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"
par ::"[('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)})"
ML {* use_legacy_bindings (the_context ()) *}
end