SSL CompoExecs.thy Interaktion und
PortierbarkeitIsabelle

TitleIOA.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"
  hereProjB=(sndex \<cdot> (snd ex))"

definition Filter_ex2 :: "'a signature \ ('a, 's) pairs \ ('a, 's) pairs"
   "Filter_ex2 sig= Filter(x. fst x \ actions sig)"

definition Filter_ex
whereFilter_exfstFilter_ex2\<cdot> (snd ex))"

definition stutter2 :java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(
    (fix \<cdot>
     (LAM h ex
        (<>s.
          case ex
            nil
          |x# xs
              (flift1
                (\<lambda>p.
(If (fst\<> actions sig Def (s=snd else)
                  andalso (h\<cdot>xs) (snd p)) \<cdot> x))))"

definition               (flift1
  where " 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"
  wherepar_execsExecsB
    (let
      exA = fstjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      exB = fst ExecsB;   "stutter sig \ (stutter2 sig \ (snd ex)) (fst ex) \ FF"
     in
       ({ex. Filter_ex sigA (ProjA ex) \<in> exA} \<inter>
        exFilter_ex (ProjBex
        {x stutter sigAProjA)} \<inter>
        {ex. stutter sigB    (let
        {. Forall(\<lambda>x. fst x \<in> actions sigA \<union> actions sigB) (snd ex)},
        asig_comp       exBfst; sigB=snd

s [ del]  split_paired_All({exFilter_ex (ProjAex

\<open>Recursive equations of operators\<close>

subsection \<open>\<open>ProjA2\<close>\<close>

lemma ProjA2_UU        exstutter sigBProjB)} \<inter>
  by( add:ProjA2_def

lemma ProjA2_nil sigA sigB"
  by

lemma ProjA2_cons: "ProjA2 \ ((a, t) \ xs) = (a, fst t) \ ProjA2 \ xs"
  by (imp: ProjA2_def

subsection

lemma ProjB2_UU:
  by simp: ProjB2_def

lemma ProjB2_nil: "ProjB2 \ nil = nil"
  by (simp

lemma ProjB2_cons: "ProjB2 \ ((a, t) \ xs) = (a, snd t) \ ProjB2 \ xs"
  by   by (simp add: ProjA2_def

subsection

lemma Filter_ex2_UUFilter_ex2 <cdot> UU = UU"
  by (simp add: Filter_ex2_def)

lemma Filter_ex2_nil: "Filter_ex2 sig \ nil = nil"

lemma Filter_ex2_cons:
  "Filter_ex2 sig \ (at \ xs) =
java.lang.StringIndexOutOfBoundsException: Range [14, 4) out of bounds for length 32
     then add)
     else
add: Filter_ex2_def

subsection \<open>\<open>stutter2\<close>\<close>

lemma stutter2_unfold
  "stutter2 siglemma : "Filter_ex2sig
    (LAM ex.
      (\<lambda>s.)
        :
          nil
         "Filter_ex2 sig\ (at \ xs) =
            (flift1
              (\<lambda>p.
                (If fst\notin actions) thenDef(=snd else)
                andalso
  apply trans
  apply (rulestutter2
ply only stutter2_def)
        (\<lambda>s.
  apply(simp add)
  done

lemma stutter2_UU           \<Rightarrow> TT
  applyjava.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 81
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

lemma rule
apply( stutter2_unfoldjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
apply
  apply stutter2_unfold

lemma:
" \cdot at \ xs)) s =
    ifat
      andalso (stutter2 sig
  apply rule)
  applyapply( stutter2_unfold
  applysimp: Consq_def )
  apply simp
done

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

subsection

lemma
  by (simp: java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

lemma stutter_nil: "stutter sig (s, nil)"
tter_def

lemma stutter_consstutter,)\<
  "stutter sig by( add: stutter_def)
  by (declare  [simp]  [simp] stutter2_cons ]

declare compoex_simps=ProjA2_UU ProjA2_cons

lemmas compoex_simps = ProjA2_UU ProjA2_nil ProjA2_cons
  ProjB2_UU ProjB2_nil ProjB2_cons
    ProjB2_UU ProjB2_cons
stutter_UU stutter_cons

declare compoex_simps [simp]

section

lemma
  section \<open>Compositionality on execution level\<close>
  "\s. is_exec_frag (A \ B) (s, xs) \
   is_exec_frag fst (asig_of
    is_exec_frag B (snd s, Filter_ex2  <comment> \<open>\<open>is_ex_fr\<close> propagates from \<open>A \<parallel> B\<close> to projections \<open>A\<close> and \<open>B\<close>\<close>
  apply (pair_induct xs simp: is_exec_frag_def Bsnd (asig_of
  text ( xs simp)
  apply (auto
  done

lemmalemma_1_1b
  \<comment> \<open>\<open>is_ex_fr (A \<parallel> B)\<close> implies stuttering on projections\<close>
:
    stutter

  apply (pair_induct pair_induct simp sforall_def)
\<open>main case\<close>
  apply (auto simp
  done

lemma
  \<comment> \<open>Executions of \<open>A \<parallel> B\<close> have only \<open>A\<close>- or \<open>B\<close>-actions\<close>

  apply (pair_induct xs : Forall_def )
  text \<open>main case\<close>
  apply auto
  apply (simp add: trans_of_defs2   \<forall>s.
  done

lemma lemma_1_2:
  \<comment> \<open>\<open>ex A\<close>, \<open>exB\<close>, stuttering and forall \<open>a \<in> A \<parallel> B\<close> implies \<open>ex (A \<parallel> B)\<close>\<close> )java.lang.NullPointerException
  "\s.
     (\<lambda>x. fst x \<in> act (A \<parallel> B)) xs \<longrightarrow>(
    is_exec_frag s  (asig_of
    stutter (asig_of A) (fst s, ProjA2 auto:  actions_asig_comp)
    stutter
Forall
    is_exec_frag (asig_of ProjA) \<in> executions A \<and>
mp sforall_defstutter_def
  apply (autostutter )( ex
  done

theorem:
  " (:executions_def ProjB_def Filter_ex_def starts_of_par)
Filter_ex A ( ex
    Filter_ex (asig_of B) (ProjB rule)
    stutter (asig_of A) (ProjA ex) \<and>
    stutter B) ( ex\<and>
    Forall (\<lambda>x. fst x \<in> act (A \<parallel> B)) (snd ex)"
   ( add ProjB_def ProjA_def)
  apply (pair extext
  apply ( apply( conjE
   \<open>\<open>\<Longrightarrow>\<close>\<close>
  apply (erule conjE)+
  apply (simp add: lemma_1_1a lemma_1_1b lemma_1_1c)
  text
  apply (erule conjE)+
  apply    simp actions_of_par
  done

theorem compositionality_ex_modules: "Execs (A \ 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

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SSL CompoExecs.thy Interaktion und
PortierbarkeitIsabelle