Theory CompoExecs

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theory CompoExecs
imports Traces
uses [CompoExecs.ML]
begin

(*  Title:      HOLCF/IOA/meta_theory/CompoExecs.thy
    ID:         $Id: CompoExecs.thy,v 1.8 2005/09/02 15:24:00 wenzelm Exp $
    Author:     Olaf Müller
*)

header {* Compositionality on Execution level *}

theory CompoExecs
imports Traces
begin

consts

 ProjA      ::"('a,'s * 't)execution => ('a,'s)execution"
 ProjA2     ::"('a,'s * 't)pairs     -> ('a,'s)pairs"
 ProjB      ::"('a,'s * 't)execution => ('a,'t)execution"
 ProjB2     ::"('a,'s * 't)pairs     -> ('a,'t)pairs"
 Filter_ex  ::"'a signature => ('a,'s)execution => ('a,'s)execution"
 Filter_ex2 ::"'a signature => ('a,'s)pairs     -> ('a,'s)pairs"
 stutter    ::"'a signature => ('a,'s)execution => bool"
 stutter2   ::"'a signature => ('a,'s)pairs     -> ('s => tr)"

 par_execs  ::"[('a,'s)execution_module,('a,'t)execution_module] => ('a,'s*'t)execution_module"


defs


ProjA_def:
 "ProjA ex == (fst (fst ex), ProjA2$(snd ex))"

ProjB_def:
 "ProjB ex == (snd (fst ex), ProjB2$(snd ex))"


ProjA2_def:
  "ProjA2 == Map (%x.(fst x,fst(snd x)))"

ProjB2_def:
  "ProjB2 == Map (%x.(fst x,snd(snd x)))"


Filter_ex_def:
  "Filter_ex sig ex == (fst ex,Filter_ex2 sig$(snd ex))"


Filter_ex2_def:
  "Filter_ex2 sig ==  Filter (%x. fst x:actions sig)"

stutter_def:
  "stutter sig ex == ((stutter2 sig$(snd ex)) (fst ex) ~= FF)"

stutter2_def:
  "stutter2 sig ==(fix$(LAM h ex. (%s. case ex of
      nil => TT
    | x##xs => (flift1
            (%p.(If Def ((fst p)~:actions sig)
                 then Def (s=(snd p))
                 else TT fi)
                andalso (h$xs) (snd p))
             $x)
   )))"

par_execs_def:
  "par_execs ExecsA ExecsB ==
       let exA = fst ExecsA; sigA = snd ExecsA;
           exB = fst ExecsB; sigB = snd ExecsB
       in
       (    {ex. Filter_ex sigA (ProjA ex) : exA}
        Int {ex. Filter_ex sigB (ProjB ex) : exB}
        Int {ex. stutter sigA (ProjA ex)}
        Int {ex. stutter sigB (ProjB ex)}
        Int {ex. Forall (%x. fst x:(actions sigA Un actions sigB)) (snd ex)},
        asig_comp sigA sigB)"

ML {* use_legacy_bindings (the_context ()) *}

end

recursive equations of operators

theorem ProjA2_UU:

  ProjA2·UU = UU

theorem ProjA2_nil:

  ProjA2·nil = nil

theorem ProjA2_cons:

  ProjA2·((a, t)>>xs) = (a, fst t)>>ProjA2·xs

theorem ProjB2_UU:

  ProjB2·UU = UU

theorem ProjB2_nil:

  ProjB2·nil = nil

theorem ProjB2_cons:

  ProjB2·((a, t)>>xs) = (a, snd t)>>ProjB2·xs

theorem Filter_ex2_UU:

  Filter_ex2 sig·UU = UU

theorem Filter_ex2_nil:

  Filter_ex2 sig·nil = nil

theorem Filter_ex2_cons:

  Filter_ex2 sig·(at>>xs) =
  (if fst at ∈ actions sig then at>>Filter_ex2 sig·xs else Filter_ex2 sig·xs)

theorem stutter2_unfold:

  stutter2 sig =
  (LAM ex.
      (%s. case ex of nil => TT
           | x ## xs =>
               (FLIFT p.
                   If Def (fst p ∉ actions sig)
                      then Def (s = snd p)
                      else TT fi andalso (stutter2 sig·xs) (snd p))·x))

theorem stutter2_UU:

  (stutter2 sig·UU) s = UU

theorem stutter2_nil:

  (stutter2 sig·nil) s = TT

theorem stutter2_cons:

  (stutter2 sig·(at>>xs)) s =
  ((if fst at ∉ actions sig then Def (s = snd at)
    else TT) andalso (stutter2 sig·xs) (snd at))

theorem stutter_UU:

  stutter sig (s, UU)

theorem stutter_nil:

  stutter sig (s, nil)

theorem stutter_cons:

  stutter sig (s, (a, t)>>ex) =
  ((a ∉ actions sig --> s = t) ∧ stutter sig (t, ex))

theorem lemma_1_1a:

s. is_exec_frag (A || B) (s, xs) -->
      is_exec_frag A (fst s, Filter_ex2 (asig_of A)·(ProjA2·xs)) ∧
      is_exec_frag B (snd s, Filter_ex2 (asig_of B)·(ProjB2·xs))

theorem lemma_1_1b:

s. is_exec_frag (A || B) (s, xs) -->
      stutter (asig_of A) (fst s, ProjA2·xs) ∧
      stutter (asig_of B) (snd s, ProjB2·xs)

theorem lemma_1_1c:

s. is_exec_frag (A || B) (s, xs) --> Forall (%x. fst x ∈ act (A || B)) xs

theorem lemma_1_2:

s. is_exec_frag A (fst s, Filter_ex2 (asig_of A)·(ProjA2·xs)) ∧
      is_exec_frag B (snd s, Filter_ex2 (asig_of B)·(ProjB2·xs)) ∧
      stutter (asig_of A) (fst s, ProjA2·xs) ∧
      stutter (asig_of B) (snd s, ProjB2·xs) ∧
      Forall (%x. fst x ∈ act (A || B)) xs -->
      is_exec_frag (A || B) (s, xs)

theorem compositionality_ex:

  (ex ∈ executions (A || B)) =
  (Filter_ex (asig_of A) (ProjA ex) ∈ executions A ∧
   Filter_ex (asig_of B) (ProjB ex) ∈ executions B ∧
   stutter (asig_of A) (ProjA ex) ∧
   stutter (asig_of B) (ProjB ex) ∧ Forall (%x. fst x ∈ act (A || B)) (snd ex))

theorem compositionality_ex_modules:

  Execs (A || B) = par_execs (Execs A) (Execs B)