SSL CompoExecs.thy Sprache: Isabelle

(*  Title:      HOL/HOLCF/IOA/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(*  :      HOL/HOLCF/IOA/CompoExecsjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44

definitionw " ex snd (fst ), ProjB2java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
where   java.lang.NullPointerException

   " sig ex = ( ex, sig
  where "

definition stutter2 :: "'a signature \ ('a, 's) pairs \ ('s \ tr)"
  where "stutter2 sig =
    fix\<cdot>
      (LAM.
(lambda of
          case \<Rightarrow> TT  # \<Rightarrow>
                   Def p <otin) then   p)  TT
          | x ## xs \<Rightarrow>
              flift1
                (\<lambda>p.wherestutter
"a'  Rightarrowt
                  andalso " ExecsA =

definition stutter :: "'a signature \ ('a, 's) execution \ bool"
whereex

definition par_execs ::
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  where {.  sigB ) \<in> exB} \<inter>{.stutter ( ex
    let
      exA ex (\<lambda>x. fst x \<in> actions sigA \<union> actions sigB) (snd ex)},
       =  ExecsB   ExecsB
lemma simp=
       {.  sigA ) \<in> exA} \<inter>

        {exjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
{. stutter ( ex
        {ex. Forall    simp )
        asig_comp))"

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

sectionby( add)

\<open>\<open>ProjB2\<close>\<close>

lemma( add)
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
simp)

lemma
   \<open>\<open>Filter_ex2\<close>\<close>lemma: " sig\java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma     (if fst at \<in> actions sig
  by (simp add: ProjB2_def)

lemma ProjB2_nil: "ProjB2 \ nil = nil"
  by (simp add: ProjB2_def)

lemma ProjB2_cons: "ProjB2 \ ((a, t) \ xs) = (a, snd t) \ ProjB2 \ xs"
  by (simp: ProjB2_def

  by (simp)

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (simp

Filter_ex2_nil  \<cdot> nil = nil"
lter_ex2_def

lter_ex2_cons \<Rightarrow> TT

    (if fst at
     then
     else Filter_ex2 Def( p <> sig  s  p)  TT
  by (simp add: Filter_ex2_def)

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

lemma stutter2_unfold   (rule)
  " sig =
     (simp:stutter2_def
(
        case ex of simp: flift1_def
nil
        | x ## xs \<Rightarrow>
            (flift1
              (\<lambda>p.
                (If Def (fst p \<notin> actions sig) then Def (s= snd p) else TT)
                andalso  applysimp
  apply (rule trans)
  apply (rule fix_eq2)
  apply (simp only: stutter2_def  done
  apply ( beta_cfun)
  apply    (subst)
  done

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

  apply simp
   (subst)
  apply simp
  done

lemma stutter2_cons stutter2_cons
  " (stutter2sig (at \ xs)) s =
(( fst  \<notin> actions sig then Def (s = snd at) else TT)
      andalso(stutter2 sig \<cdot> xs) (snd at))"
  apply(rule trans
   (ubst)
  apply (simp add (simp add flift1_defIf_and_if
  apply  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

declare

subsection addstutter_def)

lemma stutter_UU: "stutter sig (s, UU)"
  by (simp addjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma stutter_nil: tter_def)


lemma:
  "stutter sig (s, (a,t ex) \ (a \ actions sig \ (s = t)) \ stutter sig (t, ex)"
   simp

declare stutter2_UU del stutter2_nil del [simpdel

lemmas = ProjA2_niljava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
ProjB2_nil
     stutter_nilstutter_cons
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

declare

java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

lemma lemma_1_1a:is_exec_frag A( s, Filter_ex2 A) \<cdot> (ProjA2 \<cdot> xs)) \<and>
java.lang.StringIndexOutOfBoundsException: Index 151 out of bounds for length 151
  "\s. is_exec_frag (A \ B) (s, xs) \
    is_exec_frag A (fst s, Filter_ex2 (asig_of A) \<cdot> (ProjA2 \<cdot> xs)) \<and>
    is_exec_frag ( s, Filter_ex2 B) \<cdot> (ProjB2 \<cdot> xs))"
  apply pair_induct : is_exec_frag_def
  text \<open>main case\<close>
  applylemma lemma_1_1b:
  done

lemma lemma_1_1b:
  \<comment> \<open>\<open>is_ex_fr (A \<parallel> B)\<close> implies stuttering on projections\<close>
  "\s. is_exec_frag (A \ B) (s, xs) \
    stutter (asig_of A) (fst s, ProjA2 \<cdot> xs) \<and>
    stutter (asig_of B) (snd s, ProjB2 \<cdot> xs)"
  apply (pair_induct xs simp: stutter_def is_exec_frag_def)
  text \<open>main case\<close>
  apply (auto simp add: trans_of_defs2)
  done

lemma lemma_1_1c
> \<open>Executions of \<open>A \<parallel> B\<close> have only \<open>A\<close>- or \<open>B\<close>-actions\<close>
"\s. is_exec_frag (A \ B) (s, xs) \ Forall (\x. fst x \ act (A \ B)) xs"
  apply (pair_induct xs: Forall_def is_exec_frag_def
  text \<open>main case\<close>  text
  apply auto
  apply (simp
  done

lemma lemma_1_2\<comment> \<open>Executions of \<open>A \<parallel> B\<close> have only \<open>A\<close>- or \<open>B\<close>-actions\<close>
  \<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> simp sforall_defis_exec_frag_def
  "
    is_exec_frag A (fst  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    is_exec_frag B (snd s, Filter_ex2 (asig_of B <cdot> (ProjB2 \<cdot> xs)) \<and>
    stutter (asig_of A) (fst s, ProjA2 \<cdot> xs) \<and>
    stutter
Forall
    is_exec_frag A \<parallel> B) (s, xs)"
  apply (pair_induct xs simp: Forall_def sforall_def is_exec_frag_def B (snd,Filter_ex2 B) \<cdot> (ProjB2 \<cdot> xs)) \<and>
  apply( simp addtrans_of_defs1 asig_of_par
  done

positionality_ex:
  "ex (\x. fst x \ act (A \ B)) xs \
    Filter_ex A)(ProjA ex
    Filter_ex (asig_of B)    Filter_ex (asig_of B) : Forall_def is_exec_frag_def )
     (asig_ofA (rojA) \<and>
    stutter
    Forall compositionality_ex
  applysimp add executions_defProjA_def
     (asig_of) (ProjA) \<in> executions A \<and>
  apply( iffI)
  text \<open>\<open>\<Longrightarrow>\<close>\<close> (asig_ofProjB)
  apply (erule conjE)+
  apply (simp  apply simp: executions_def Filter_ex_def starts_of_par
  text \<open>\<open>\<Longleftarrow>\<close>\<close>
apply erule)+
  apply (simp add: lemma_1_2)
  done

theorem  text
  apply (unfold java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  apply (simp add: asig_of_par
  apply (rule set_eqItext \<open>\<open>\<Longleftarrow>\<close>\<close>
  apply( add: compositionality_ex)
  done

end

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