src/HOL/HOLCF/IOA/meta_theory/CompoExecs.thy

isabelle update_cartouches -c -t;

(*  Title:      HOL/HOLCF/IOA/meta_theory/CompoExecs.thy
    Author:     Olaf Müller
*)

section \<open>Compositionality on Execution level\<close>

theory CompoExecs
imports Traces
begin

definition
  ProjA2 :: "('a,'s * 't)pairs -> ('a,'s)pairs" where
  "ProjA2 = Map (%x.(fst x,fst(snd x)))"

definition
  ProjA :: "('a,'s * 't)execution => ('a,'s)execution" where
  "ProjA ex = (fst (fst ex), ProjA2$(snd ex))"

definition
  ProjB2 :: "('a,'s * 't)pairs -> ('a,'t)pairs" where
  "ProjB2 = Map (%x.(fst x,snd(snd x)))"

definition
  ProjB :: "('a,'s * 't)execution => ('a,'t)execution" where
  "ProjB ex = (snd (fst ex), ProjB2$(snd ex))"

definition
  Filter_ex2 :: "'a signature => ('a,'s)pairs -> ('a,'s)pairs" where
  "Filter_ex2 sig = Filter (%x. fst x:actions sig)"

definition
  Filter_ex :: "'a signature => ('a,'s)execution => ('a,'s)execution" where
  "Filter_ex sig ex = (fst ex,Filter_ex2 sig$(snd ex))"

definition
  stutter2 :: "'a signature => ('a,'s)pairs -> ('s => tr)" where
  "stutter2 sig = (fix$(LAM h ex. (%s. case ex of
      nil => TT
    | x##xs => (flift1
            (%p.(If Def ((fst p)~:actions sig)
                 then Def (s=(snd p))
                 else TT)
                andalso (h$xs) (snd p))
             $x)
   )))"

definition
  stutter :: "'a signature => ('a,'s)execution => bool" where
  "stutter sig ex = ((stutter2 sig$(snd ex)) (fst ex) ~= FF)"

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


lemmas [simp del] = split_paired_All


section "recursive equations of operators"


(* ---------------------------------------------------------------- *)
(*                               ProjA2                             *)
(* ---------------------------------------------------------------- *)


lemma ProjA2_UU: "ProjA2$UU = UU"
apply (simp add: ProjA2_def)
done

lemma ProjA2_nil: "ProjA2$nil = nil"
apply (simp add: ProjA2_def)
done

lemma ProjA2_cons: "ProjA2$((a,t)\<leadsto>xs) = (a,fst t) \<leadsto> ProjA2$xs"
apply (simp add: ProjA2_def)
done


(* ---------------------------------------------------------------- *)
(*                               ProjB2                             *)
(* ---------------------------------------------------------------- *)


lemma ProjB2_UU: "ProjB2$UU = UU"
apply (simp add: ProjB2_def)
done

lemma ProjB2_nil: "ProjB2$nil = nil"
apply (simp add: ProjB2_def)
done

lemma ProjB2_cons: "ProjB2$((a,t)\<leadsto>xs) = (a,snd t) \<leadsto> ProjB2$xs"
apply (simp add: ProjB2_def)
done



(* ---------------------------------------------------------------- *)
(*                             Filter_ex2                           *)
(* ---------------------------------------------------------------- *)


lemma Filter_ex2_UU: "Filter_ex2 sig$UU=UU"
apply (simp add: Filter_ex2_def)
done

lemma Filter_ex2_nil: "Filter_ex2 sig$nil=nil"
apply (simp add: Filter_ex2_def)
done

lemma Filter_ex2_cons: "Filter_ex2 sig$(at \<leadsto> xs) =
             (if (fst at:actions sig)
                  then at \<leadsto> (Filter_ex2 sig$xs)
                  else        Filter_ex2 sig$xs)"

apply (simp add: Filter_ex2_def)
done


(* ---------------------------------------------------------------- *)
(*                             stutter2                             *)
(* ---------------------------------------------------------------- *)


lemma stutter2_unfold: "stutter2 sig = (LAM ex. (%s. case ex of
       nil => TT
     | x##xs => (flift1
             (%p.(If Def ((fst p)~:actions sig)
                  then Def (s=(snd p))
                  else TT)
                 andalso (stutter2 sig$xs) (snd p))
              $x)
            ))"
apply (rule trans)
apply (rule fix_eq2)
apply (simp only: stutter2_def)
apply (rule beta_cfun)
apply (simp add: flift1_def)
done

lemma stutter2_UU: "(stutter2 sig$UU) s=UU"
apply (subst stutter2_unfold)
apply simp
done

lemma stutter2_nil: "(stutter2 sig$nil) s = TT"
apply (subst stutter2_unfold)
apply simp
done

lemma stutter2_cons: "(stutter2 sig$(at\<leadsto>xs)) s =
               ((if (fst at)~:actions sig then Def (s=snd at) else TT)
                 andalso (stutter2 sig$xs) (snd at))"
apply (rule trans)
apply (subst stutter2_unfold)
apply (simp add: Consq_def flift1_def If_and_if)
apply simp
done


declare stutter2_UU [simp] stutter2_nil [simp] stutter2_cons [simp]


(* ---------------------------------------------------------------- *)
(*                             stutter                              *)
(* ---------------------------------------------------------------- *)

lemma stutter_UU: "stutter sig (s, UU)"
apply (simp add: stutter_def)
done

lemma stutter_nil: "stutter sig (s, nil)"
apply (simp add: stutter_def)
done

lemma stutter_cons: "stutter sig (s, (a,t)\<leadsto>ex) =
      ((a~:actions sig --> (s=t)) & stutter sig (t,ex))"
apply (simp add: stutter_def)
done

(* ----------------------------------------------------------------------------------- *)

declare stutter2_UU [simp del] stutter2_nil [simp del] stutter2_cons [simp del]

lemmas compoex_simps = ProjA2_UU ProjA2_nil ProjA2_cons
  ProjB2_UU ProjB2_nil ProjB2_cons
  Filter_ex2_UU Filter_ex2_nil Filter_ex2_cons
  stutter_UU stutter_nil stutter_cons

declare compoex_simps [simp]



(* ------------------------------------------------------------------ *)
(*                      The following lemmata aim for                 *)
(*             COMPOSITIONALITY   on    EXECUTION     Level           *)
(* ------------------------------------------------------------------ *)


(* --------------------------------------------------------------------- *)
(*  Lemma_1_1a : is_ex_fr propagates from A\<parallel>B to Projections A and B    *)
(* --------------------------------------------------------------------- *)

lemma lemma_1_1a: "!s. is_exec_frag (A\<parallel>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))"

apply (tactic \<open>pair_induct_tac @{context} "xs" [@{thm is_exec_frag_def}] 1\<close>)
(* main case *)
apply (auto simp add: trans_of_defs2)
done


(* --------------------------------------------------------------------- *)
(*  Lemma_1_1b : is_ex_fr (A\<parallel>B) implies stuttering on Projections       *)
(* --------------------------------------------------------------------- *)

lemma lemma_1_1b: "!s. is_exec_frag (A\<parallel>B) (s,xs)
       --> stutter (asig_of A) (fst s,ProjA2$xs)  &
           stutter (asig_of B) (snd s,ProjB2$xs)"

apply (tactic \<open>pair_induct_tac @{context} "xs"
  [@{thm stutter_def}, @{thm is_exec_frag_def}] 1\<close>)
(* main case *)
apply (auto simp add: trans_of_defs2)
done


(* --------------------------------------------------------------------- *)
(*  Lemma_1_1c : Executions of A\<parallel>B have only  A- or B-actions           *)
(* --------------------------------------------------------------------- *)

lemma lemma_1_1c: "!s. (is_exec_frag (A\<parallel>B) (s,xs)
   --> Forall (%x. fst x:act (A\<parallel>B)) xs)"

apply (tactic \<open>pair_induct_tac @{context} "xs" [@{thm Forall_def}, @{thm sforall_def},
  @{thm is_exec_frag_def}] 1\<close>)
(* main case *)
apply auto
apply (simp add: trans_of_defs2 actions_asig_comp asig_of_par)
done


(* ----------------------------------------------------------------------- *)
(*  Lemma_1_2 : ex A, exB, stuttering and forall a:A|B implies ex (A\<parallel>B)   *)
(* ----------------------------------------------------------------------- *)

lemma 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\<parallel>B)) xs
     --> is_exec_frag (A\<parallel>B) (s,xs)"

apply (tactic \<open>pair_induct_tac @{context} "xs" [@{thm Forall_def}, @{thm sforall_def},
  @{thm is_exec_frag_def}, @{thm stutter_def}] 1\<close>)
apply (auto simp add: trans_of_defs1 actions_asig_comp asig_of_par)
done


subsection \<open>COMPOSITIONALITY on EXECUTION Level -- Main Theorem\<close>

lemma compositionality_ex:
"(ex:executions(A\<parallel>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\<parallel>B)) (snd ex))"

apply (simp (no_asm) add: executions_def ProjB_def Filter_ex_def ProjA_def starts_of_par)
apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
apply (rule iffI)
(* ==>  *)
apply (erule conjE)+
apply (simp add: lemma_1_1a lemma_1_1b lemma_1_1c)
(* <==  *)
apply (erule conjE)+
apply (simp add: lemma_1_2)
done


subsection \<open>COMPOSITIONALITY on EXECUTION Level -- for Modules\<close>

lemma compositionality_ex_modules:
  "Execs (A\<parallel>B) = par_execs (Execs A) (Execs B)"
apply (unfold Execs_def par_execs_def)
apply (simp add: asig_of_par)
apply (rule set_eqI)
apply (simp add: compositionality_ex actions_of_par)
done

end