# Theory IOA

```(*  Title:      HOL/IOA/IOA.thy
Author:     Tobias Nipkow & Konrad Slind
Copyright   1994  TU Muenchen
*)

section ‹The I/O automata of Lynch and Tuttle›

theory IOA
imports Asig
begin

type_synonym 'a seq = "nat => 'a"
type_synonym 'a oseq = "nat => 'a option"
type_synonym ('a, 'b) execution = "'a oseq * 'b seq"
type_synonym ('a, 's) transition = "('s * 'a * 's)"
type_synonym ('a,'s) ioa = "'a signature * 's set * ('a, 's) transition set"

(* IO automata *)

definition state_trans :: "['action signature, ('action,'state)transition set] => bool"
where "state_trans asig R ≡
(∀triple. triple ∈ R ⟶ fst(snd(triple)) ∈ actions(asig)) ∧
(∀a. (a ∈ inputs(asig)) ⟶ (∀s1. ∃s2. (s1,a,s2) ∈ R))"

definition asig_of :: "('action,'state)ioa => 'action signature"
where "asig_of == fst"

definition starts_of :: "('action,'state)ioa => 'state set"
where "starts_of == (fst o snd)"

definition trans_of :: "('action,'state)ioa => ('action,'state)transition set"
where "trans_of == (snd o snd)"

definition IOA :: "('action,'state)ioa => bool"
where "IOA(ioa) == (is_asig(asig_of(ioa)) &
(~ starts_of(ioa) = {}) &
state_trans (asig_of ioa) (trans_of ioa))"

(* Executions, schedules, and traces *)

(* 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. *)
definition is_execution_fragment :: "[('action,'state)ioa, ('action,'state)execution] => bool"
where "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))"

definition executions :: "('action,'state)ioa => ('action,'state)execution set"
where "executions(ioa) ≡ {e. snd e 0 ∈ starts_of(ioa) ∧ is_execution_fragment ioa e}"

definition reachable :: "[('action,'state)ioa, 'state] => bool"
where "reachable ioa s ≡ (∃ex∈executions(ioa). ∃n. (snd ex n) = s)"

definition invariant :: "[('action,'state)ioa, 'state=>bool] => bool"
where "invariant A P ≡ (∀s. reachable A s ⟶ P(s))"

(* Composition of action signatures and automata *)

consts
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 *)

definition compat_asigs ::"['action signature, 'action signature] => bool"
where "compat_asigs a1 a2 ==
(((outputs(a1) Int outputs(a2)) = {}) ∧
((internals(a1) Int actions(a2)) = {}) ∧
((internals(a2) Int actions(a1)) = {}))"

definition compat_ioas  ::"[('action,'s)ioa, ('action,'t)ioa] ⇒ bool"
where "compat_ioas ioa1 ioa2 ≡ compat_asigs (asig_of(ioa1)) (asig_of(ioa2))"

definition asig_comp :: "['action signature, 'action signature] ⇒ 'action signature"
where "asig_comp a1 a2 ≡
(((inputs(a1) ∪ inputs(a2)) - (outputs(a1) ∪ outputs(a2)),
(outputs(a1) ∪ outputs(a2)),
(internals(a1) ∪ internals(a2))))"

definition par :: "[('a,'s)ioa, ('a,'t)ioa] ⇒ ('a,'s*'t)ioa"  (infixr "||" 10)
where "(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))})"

(* Filtering and hiding *)

(* Restrict the trace to those members of the set s *)
definition filter_oseq :: "('a => bool) => 'a oseq => 'a oseq"
where "filter_oseq p s ≡
(λi. case s(i)
of None ⇒ None
| Some(x) ⇒ if p x then Some x else None)"

definition mk_trace :: "[('action,'state)ioa, 'action oseq] ⇒ 'action oseq"
where "mk_trace(ioa) ≡ filter_oseq(λa. a ∈ externals(asig_of(ioa)))"

(* Does an ioa have an execution with the given trace *)
definition has_trace :: "[('action,'state)ioa, 'action oseq] ⇒ bool"
where "has_trace ioa b ≡ (∃ex∈executions(ioa). b = mk_trace ioa (fst ex))"

definition NF :: "'a oseq => 'a oseq"
where "NF(tr) ≡ SOME 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 *)
definition traces :: "('action,'state)ioa => 'action oseq set"
where "traces(ioa) ≡ {trace. ∃tr. trace=NF(tr) ∧ has_trace ioa tr}"

definition restrict_asig :: "['a signature, 'a set] => 'a signature"
where "restrict_asig asig actns ≡
(inputs(asig) ∩ actns, outputs(asig) ∩ actns,
internals(asig) ∪ (externals(asig) - actns))"

definition restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
where "restrict ioa actns ≡
(restrict_asig (asig_of ioa) actns, starts_of(ioa), trans_of(ioa))"

(* Notions of correctness *)

definition ioa_implements :: "[('action,'state1)ioa, ('action,'state2)ioa] => bool"
where "ioa_implements ioa1 ioa2 ≡
((inputs(asig_of(ioa1)) = inputs(asig_of(ioa2))) ∧
(outputs(asig_of(ioa1)) = outputs(asig_of(ioa2))) ∧
traces(ioa1) ⊆ traces(ioa2))"

(* Instantiation of abstract IOA by concrete actions *)

definition rename :: "('a, 'b)ioa ⇒ ('c ⇒ 'a option) ⇒ ('c,'b)ioa"
where "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) ∪ 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 prod_eq_iff 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)) ∪ 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
```