Theory IOA (Isabelle2005: October 2005)

(*  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)