(* Title: HOL/IOA/IOA.thy ID: $Id: IOA.thy,v 1.5 2005/09/06 17:03:39 wenzelm Exp $ 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
theorems ioa_projections:
asig_of == fst
starts_of == fst o snd
trans_of == snd o snd
theorems exec_rws:
executions ioa == {e. snd e 0 ∈ starts_of ioa ∧ is_execution_fragment ioa e}
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)
theorem ioa_triple_proj:
asig_of (x, y, z) = x ∧ starts_of (x, y, z) = y ∧ trans_of (x, y, z) = z
theorem trans_in_actions:
[| IOA A; (s1.0, a, s2.0) ∈ trans_of A |] ==> a ∈ actions (asig_of A)
theorem filter_oseq_idemp:
filter_oseq p (filter_oseq p s) = filter_oseq p s
theorem mk_trace_thm:
(mk_trace A s n = None) = (s n = None ∨ (∃a. s n = Some a ∧ a ∉ externals (asig_of A))) ∧ (mk_trace A s n = Some a) = (s n = Some a ∧ a ∈ externals (asig_of A))
theorem reachable_0:
s ∈ starts_of A ==> reachable A s
theorem reachable_n:
[| reachable A s; (s, a, t) ∈ trans_of A |] ==> reachable A t
theorem invariantI:
[| !!s. s ∈ starts_of A ==> P s; !!s t a. [| reachable A s; P s |] ==> (s, a, t) ∈ trans_of A --> P t |] ==> invariant A P
theorem invariantI1:
[| !!s. s ∈ starts_of A ==> P s; !!s t a. reachable A s ==> P s --> (s, a, t) ∈ trans_of A --> P t |] ==> invariant A P
theorem invariantE:
[| invariant A P; reachable A s |] ==> P s
theorem actions_asig_comp:
actions (asig_comp a b) = actions a ∪ actions b
theorem starts_of_par:
starts_of (A || B) = {p. fst p ∈ starts_of A ∧ snd p ∈ starts_of B}
theorem states_of_exec_reachable:
ex ∈ executions A ==> ∀n. reachable A (snd ex n)
theorem trans_of_par4:
((s, a, t) ∈ trans_of (A || B || C || D)) = ((a ∈ actions (asig_of A) ∨ a ∈ actions (asig_of B) ∨ a ∈ actions (asig_of C) ∨ a ∈ actions (asig_of D)) ∧ (if a ∈ actions (asig_of A) then (fst s, a, fst t) ∈ trans_of A else fst t = fst s) ∧ (if a ∈ actions (asig_of B) then (fst (snd s), a, fst (snd t)) ∈ trans_of B else fst (snd t) = fst (snd s)) ∧ (if a ∈ actions (asig_of C) then (fst (snd (snd s)), a, fst (snd (snd t))) ∈ trans_of C else fst (snd (snd t)) = fst (snd (snd s))) ∧ (if a ∈ actions (asig_of D) then (snd (snd (snd s)), a, snd (snd (snd t))) ∈ trans_of D else snd (snd (snd t)) = snd (snd (snd s))))
theorem cancel_restrict:
starts_of (restrict ioa acts) = starts_of ioa ∧ trans_of (restrict ioa acts) = trans_of ioa ∧ reachable (restrict ioa acts) s = reachable ioa s
theorem asig_of_par:
asig_of (A || B) = asig_comp (asig_of A) (asig_of B)
theorem externals_of_par:
externals (asig_of (A1.0 || A2.0)) = externals (asig_of A1.0) ∪ externals (asig_of A2.0)
theorem ext1_is_not_int2:
[| compat_ioas A1.0 A2.0; a ∈ externals (asig_of A1.0) |] ==> a ∉ internals (asig_of A2.0)
theorem ext2_is_not_int1:
[| compat_ioas A2.0 A1.0; a ∈ externals (asig_of A1.0) |] ==> a ∉ internals (asig_of A2.0)