(* 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 \<times> 't) pairs \<rightarrow> ('a, 's) pairs"
where "ProjA2 = Map (\<lambda>x. (fst x, fst (snd x)))"
definition ProjA :: "('a, 's \<times> 't) execution \<Rightarrow> ('a, 's) execution"
where "ProjA ex = (fst (fst ex), ProjA2 \<cdot> (snd ex))"
definition ProjB2 :: "('a, 's \<times> 't) pairs \<rightarrow> ('a, 't) pairs"
where "ProjB2 = Map (\<lambda>x. (fst x, snd (snd x)))"
definition ProjB :: "('a, 's \<times> 't) execution \<Rightarrow> ('a, 't) execution"
where "ProjB ex = (snd (fst ex), ProjB2 \<cdot> (snd ex))"
definition Filter_ex2 :: "'a signature \<Rightarrow> ('a, 's) pairs \<rightarrow> ('a, 's) pairs"
where "Filter_ex2 sig = Filter (\<lambda>x. fst x \<in> actions sig)"
definition Filter_ex :: "'a signature \<Rightarrow> ('a, 's) execution \<Rightarrow> ('a, 's) execution"
where "Filter_ex sig ex = (fst ex, Filter_ex2 sig \<cdot> (snd ex))"
definition stutter2 :: "'a signature \<Rightarrow> ('a, 's) pairs \<rightarrow> ('s \<Rightarrow> tr)"
where "stutter2 sig =
(fix \<cdot>
(LAM h ex.
(\<lambda>s.
case ex of
nil \<Rightarrow> TT
| x ## xs \<Rightarrow>
(flift1
(\<lambda>p.
(If Def (fst p \<notin> actions sig) then Def (s = snd p) else TT)
andalso (h\<cdot>xs) (snd p)) \<cdot> x))))"
definition stutter :: "'a signature \<Rightarrow> ('a, 's) execution \<Rightarrow> bool"
where "stutter sig ex \<longleftrightarrow> (stutter2 sig \<cdot> (snd ex)) (fst ex) \<noteq> FF"
definition par_execs ::
"('a, 's) execution_module \<Rightarrow> ('a, 't) execution_module \<Rightarrow> ('a, 's \<times> '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) \<in> exA} \<inter>
{ex. Filter_ex sigB (ProjB ex) \<in> exB} \<inter>
{ex. stutter sigA (ProjA ex)} \<inter>
{ex. stutter sigB (ProjB ex)} \<inter>
{ex. Forall (\<lambda>x. fst x \<in> actions sigA \<union> actions sigB) (snd ex)},
asig_comp sigA sigB))"
lemmas [simp del] = split_paired_All
section \<open>Recursive equations of operators\<close>
subsection \<open>\<open>ProjA2\<close>\<close>
lemma ProjA2_UU: "ProjA2 \<cdot> UU = UU"
by (simp add: ProjA2_def)
lemma ProjA2_nil: "ProjA2 \<cdot> nil = nil"
by (simp add: ProjA2_def)
lemma ProjA2_cons: "ProjA2 \<cdot> ((a, t) \<leadsto> xs) = (a, fst t) \<leadsto> ProjA2 \<cdot> xs"
by (simp add: ProjA2_def)
subsection \<open>\<open>ProjB2\<close>\<close>
lemma ProjB2_UU: "ProjB2 \<cdot> UU = UU"
by (simp add: ProjB2_def)
lemma ProjB2_nil: "ProjB2 \<cdot> nil = nil"
by (simp add: ProjB2_def)
lemma ProjB2_cons: "ProjB2 \<cdot> ((a, t) \<leadsto> xs) = (a, snd t) \<leadsto> ProjB2 \<cdot> xs"
by (simp add: ProjB2_def)
subsection \<open>\<open>Filter_ex2\<close>\<close>
lemma Filter_ex2_UU: "Filter_ex2 sig \<cdot> UU = UU"
by (simp add: Filter_ex2_def)
lemma Filter_ex2_nil: "Filter_ex2 sig \<cdot> nil = nil"
by (simp add: Filter_ex2_def)
lemma Filter_ex2_cons:
"Filter_ex2 sig \<cdot> (at \<leadsto> xs) =
(if fst at \<in> actions sig
then at \<leadsto> (Filter_ex2 sig \<cdot> xs)
else Filter_ex2 sig \<cdot> xs)"
by (simp add: Filter_ex2_def)
subsection \<open>\<open>stutter2\<close>\<close>
lemma stutter2_unfold:
"stutter2 sig =
(LAM ex.
(\<lambda>s.
case ex of
nil \<Rightarrow> TT
| x ## xs \<Rightarrow>
(flift1
(\<lambda>p.
(If Def (fst p \<notin> actions sig) then Def (s= snd p) else TT)
andalso (stutter2 sig\<cdot>xs) (snd p)) \<cdot> 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 \<cdot> UU) s = UU"
apply (subst stutter2_unfold)
apply simp
done
lemma stutter2_nil: "(stutter2 sig \<cdot> nil) s = TT"
apply (subst stutter2_unfold)
apply simp
done
lemma stutter2_cons:
"(stutter2 sig \<cdot> (at \<leadsto> xs)) s =
((if fst at \<notin> actions sig then Def (s = snd at) else TT)
andalso (stutter2 sig \<cdot> 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]
subsection \<open>\<open>stutter\<close>\<close>
lemma stutter_UU: "stutter sig (s, UU)"
by (simp add: stutter_def)
lemma stutter_nil: "stutter sig (s, nil)"
by (simp add: stutter_def)
lemma stutter_cons:
"stutter sig (s, (a, t) \<leadsto> ex) \<longleftrightarrow> (a \<notin> actions sig \<longrightarrow> (s = t)) \<and> stutter sig (t, ex)"
by (simp add: stutter_def)
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]
section \<open>Compositionality on execution level\<close>
lemma lemma_1_1a:
\<comment> \<open>\<open>is_ex_fr\<close> propagates from \<open>A \<parallel> B\<close> to projections \<open>A\<close> and \<open>B\<close>\<close>
"\<forall>s. is_exec_frag (A \<parallel> B) (s, xs) \<longrightarrow>
is_exec_frag A (fst s, Filter_ex2 (asig_of A) \<cdot> (ProjA2 \<cdot> xs)) \<and>
is_exec_frag B (snd s, Filter_ex2 (asig_of B) \<cdot> (ProjB2 \<cdot> xs))"
apply (pair_induct xs simp: is_exec_frag_def)
text \<open>main case\<close>
apply (auto simp add: trans_of_defs2)
done
lemma lemma_1_1b:
\<comment> \<open>\<open>is_ex_fr (A \<parallel> B)\<close> implies stuttering on projections\<close>
"\<forall>s. is_exec_frag (A \<parallel> B) (s, xs) \<longrightarrow>
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:
\<comment> \<open>Executions of \<open>A \<parallel> B\<close> have only \<open>A\<close>- or \<open>B\<close>-actions\<close>
"\<forall>s. is_exec_frag (A \<parallel> B) (s, xs) \<longrightarrow> Forall (\<lambda>x. fst x \<in> act (A \<parallel> B)) xs"
apply (pair_induct xs simp: Forall_def sforall_def is_exec_frag_def)
text \<open>main case\<close>
apply auto
apply (simp add: trans_of_defs2 actions_asig_comp asig_of_par)
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>
"\<forall>s.
is_exec_frag A (fst s, Filter_ex2 (asig_of A) \<cdot> (ProjA2 \<cdot> xs)) \<and>
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 (asig_of B) (snd s, ProjB2 \<cdot> xs) \<and>
Forall (\<lambda>x. fst x \<in> act (A \<parallel> B)) xs \<longrightarrow>
is_exec_frag (A \<parallel> B) (s, xs)"
apply (pair_induct xs simp: Forall_def sforall_def is_exec_frag_def stutter_def)
apply (auto simp add: trans_of_defs1 actions_asig_comp asig_of_par)
done
theorem compositionality_ex:
"ex \<in> executions (A \<parallel> B) \<longleftrightarrow>
Filter_ex (asig_of A) (ProjA ex) \<in> executions A \<and>
Filter_ex (asig_of B) (ProjB ex) \<in> executions B \<and>
stutter (asig_of A) (ProjA ex) \<and>
stutter (asig_of B) (ProjB ex) \<and>
Forall (\<lambda>x. fst x \<in> act (A \<parallel> B)) (snd ex)"
apply (simp add: executions_def ProjB_def Filter_ex_def ProjA_def starts_of_par)
apply (pair ex)
apply (rule iffI)
text \<open>\<open>\<Longrightarrow>\<close>\<close>
apply (erule conjE)+
apply (simp add: lemma_1_1a lemma_1_1b lemma_1_1c)
text \<open>\<open>\<Longleftarrow>\<close>\<close>
apply (erule conjE)+
apply (simp add: lemma_1_2)
done
theorem 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