(* Title: HOL/IOA/IOA.thy ID: $Id: IOA.thy,v 1.8 2008/05/07 08:57:29 berghofe 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)})" declare Let_def [simp] lemmas ioa_projections = asig_of_def starts_of_def trans_of_def and exec_rws = executions_def is_execution_fragment_def lemma ioa_triple_proj: "asig_of(x,y,z) = x & starts_of(x,y,z) = y & trans_of(x,y,z) = z" apply (simp add: ioa_projections) done lemma trans_in_actions: "[| IOA(A); (s1,a,s2):trans_of(A) |] ==> a:actions(asig_of(A))" apply (unfold ioa_def state_trans_def actions_def is_asig_def) apply (erule conjE)+ apply (erule allE, erule impE, assumption) apply simp done lemma filter_oseq_idemp: "filter_oseq p (filter_oseq p s) = filter_oseq p s" apply (simp add: filter_oseq_def) apply (rule ext) apply (case_tac "s i") apply simp_all done lemma 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)))" apply (unfold mk_trace_def filter_oseq_def) apply (case_tac "s n") apply auto done lemma reachable_0: "s:starts_of(A) ==> reachable A s" apply (unfold reachable_def) apply (rule_tac x = "(%i. None, %i. s)" in bexI) apply simp apply (simp add: exec_rws) done lemma reachable_n: "!!A. [| reachable A s; (s,a,t) : trans_of(A) |] ==> reachable A t" apply (unfold reachable_def exec_rws) apply (simp del: bex_simps) apply (simp (no_asm_simp) only: split_tupled_all) apply safe apply (rename_tac ex1 ex2 n) apply (rule_tac x = "(%i. if i<n then ex1 i else (if i=n then Some a else None) , %i. if i<Suc n then ex2 i else t)" in bexI) apply (rule_tac x = "Suc n" in exI) apply (simp (no_asm)) apply simp apply (metis ioa_triple_proj less_antisym) done lemma invariantI: assumes p1: "!!s. s:starts_of(A) ==> P(s)" and p2: "!!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t)" shows "invariant A P" apply (unfold invariant_def reachable_def Let_def exec_rws) apply safe apply (rename_tac ex1 ex2 n) apply (rule_tac Q = "reachable A (ex2 n) " in conjunct1) apply simp apply (induct_tac n) apply (fast intro: p1 reachable_0) apply (erule_tac x = na in allE) apply (case_tac "ex1 na", simp_all) apply safe apply (erule p2 [THEN mp]) apply (fast dest: reachable_n)+ done lemma 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" apply (blast intro!: invariantI) done lemma invariantE: "[| invariant A P; reachable A s |] ==> P(s)" apply (unfold invariant_def) apply blast done lemma actions_asig_comp: "actions(asig_comp a b) = actions(a) Un actions(b)" apply (auto simp add: actions_def asig_comp_def asig_projections) done lemma starts_of_par: "starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}" apply (simp add: par_def ioa_projections) done (* Every state in an execution is reachable *) lemma states_of_exec_reachable: "ex:executions(A) ==> !n. reachable A (snd ex n)" apply (unfold reachable_def) apply fast done lemma 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)))))" (*SLOW*) apply (simp (no_asm) add: par_def actions_asig_comp Pair_fst_snd_eq ioa_projections) done lemma 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" apply (simp add: is_execution_fragment_def executions_def reachable_def restrict_def ioa_projections) done lemma asig_of_par: "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)" apply (simp add: par_def ioa_projections) done lemma externals_of_par: "externals(asig_of(A1||A2)) = (externals(asig_of(A1)) Un externals(asig_of(A2)))" apply (simp add: externals_def asig_of_par asig_comp_def asig_inputs_def asig_outputs_def Un_def set_diff_eq) apply blast done lemma ext1_is_not_int2: "[| compat_ioas A1 A2; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))" apply (unfold externals_def actions_def compat_ioas_def compat_asigs_def) apply auto done lemma ext2_is_not_int1: "[| compat_ioas A2 A1 ; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))" apply (unfold externals_def actions_def compat_ioas_def compat_asigs_def) apply auto done lemmas ext1_ext2_is_not_act2 = ext1_is_not_int2 [THEN int_and_ext_is_act] and ext1_ext2_is_not_act1 = ext2_is_not_int1 [THEN int_and_ext_is_act] end
lemma ioa_projections:
asig_of == fst
starts_of == fst o snd
trans_of == snd o snd
and 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)
lemma ioa_triple_proj:
asig_of (x, y, z) = x ∧ starts_of (x, y, z) = y ∧ trans_of (x, y, z) = z
lemma trans_in_actions:
[| IOA A; (s1.0, a, s2.0) ∈ trans_of A |] ==> a ∈ actions (asig_of A)
lemma filter_oseq_idemp:
filter_oseq p (filter_oseq p s) = filter_oseq p s
lemma 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))
lemma reachable_0:
s ∈ starts_of A ==> reachable A s
lemma reachable_n:
[| reachable A s; (s, a, t) ∈ trans_of A |] ==> reachable A t
lemma 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
lemma 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
lemma invariantE:
[| invariant A P; reachable A s |] ==> P s
lemma actions_asig_comp:
actions (asig_comp a b) = actions a ∪ actions b
lemma starts_of_par:
starts_of (A || B) = {p. fst p ∈ starts_of A ∧ snd p ∈ starts_of B}
lemma states_of_exec_reachable:
ex ∈ executions A ==> ∀n. reachable A (snd ex n)
lemma 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))))
lemma 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
lemma asig_of_par:
asig_of (A || B) = asig_comp (asig_of A) (asig_of B)
lemma externals_of_par:
externals (asig_of (A1.0 || A2.0)) =
externals (asig_of A1.0) ∪ externals (asig_of A2.0)
lemma ext1_is_not_int2:
[| compat_ioas A1.0 A2.0; a ∈ externals (asig_of A1.0) |]
==> a ∉ internals (asig_of A2.0)
lemma ext2_is_not_int1:
[| compat_ioas A2.0 A1.0; a ∈ externals (asig_of A1.0) |]
==> a ∉ internals (asig_of A2.0)
lemma ext1_ext2_is_not_act2:
[| compat_ioas A1.1 A2.1; a ∈ externals (asig_of A1.1);
a ∉ externals (asig_of A2.1) |]
==> a ∉ actions (asig_of A2.1)
and ext1_ext2_is_not_act1:
[| compat_ioas A2.1 A1.1; a ∈ externals (asig_of A1.1);
a ∉ externals (asig_of A2.1) |]
==> a ∉ actions (asig_of A2.1)