(* Title: HOL/HOLCF/IOA/TLS.thy
Author: Olaf Müller
*)
section \<open>Temporal Logic of Steps -- tailored for I/O automata\<close>
theory TLS
imports IOA TL
begin
default_sort type
type_synonym ('a, 's) ioa_temp = "('a option, 's) transition temporal"
type_synonym ('a, 's) step_pred = "('a option, 's) transition predicate"
type_synonym 's state_pred = "'s predicate"
definition mkfin :: "'a Seq \<Rightarrow> 'a Seq"
where "mkfin s = (if Partial s then SOME t. Finite t \<and> s = t @@ UU else s)"
definition option_lift :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a option \<Rightarrow> 'b"
where "option_lift f s y = (case y of None \<Rightarrow> s | Some x \<Rightarrow> f x)"
definition plift :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool"
(* plift is used to determine that None action is always false in
transition predicates *)
where "plift P = option_lift P False"
definition xt1 :: "'s predicate \<Rightarrow> ('a, 's) step_pred"
where "xt1 P tr = P (fst tr)"
definition xt2 :: "'a option predicate \<Rightarrow> ('a, 's) step_pred"
where "xt2 P tr = P (fst (snd tr))"
definition ex2seqC :: "('a, 's) pairs \<rightarrow> ('s \<Rightarrow> ('a option, 's) transition Seq)"
where "ex2seqC =
(fix $ (LAM h ex. (\<lambda>s. case ex of
nil \<Rightarrow> (s, None, s) \<leadsto> nil
| x ## xs \<Rightarrow> (flift1 (\<lambda>pr. (s, Some (fst pr), snd pr) \<leadsto> (h $ xs) (snd pr)) $ x))))"
definition ex2seq :: "('a, 's) execution \<Rightarrow> ('a option, 's) transition Seq"
where "ex2seq ex = (ex2seqC $ (mkfin (snd ex))) (fst ex)"
definition temp_sat :: "('a, 's) execution \<Rightarrow> ('a, 's) ioa_temp \<Rightarrow> bool" (infixr "\<TTurnstile>" 22)
where "(ex \<TTurnstile> P) \<longleftrightarrow> ((ex2seq ex) \<Turnstile> P)"
definition validTE :: "('a, 's) ioa_temp \<Rightarrow> bool"
where "validTE P \<longleftrightarrow> (\<forall>ex. (ex \<TTurnstile> P))"
definition validIOA :: "('a, 's) ioa \<Rightarrow> ('a, 's) ioa_temp \<Rightarrow> bool"
where "validIOA A P \<longleftrightarrow> (\<forall>ex \<in> executions A. (ex \<TTurnstile> P))"
lemma IMPLIES_temp_sat [simp]: "(ex \<TTurnstile> P \<^bold>\<longrightarrow> Q) \<longleftrightarrow> ((ex \<TTurnstile> P) \<longrightarrow> (ex \<TTurnstile> Q))"
by (simp add: IMPLIES_def temp_sat_def satisfies_def)
lemma AND_temp_sat [simp]: "(ex \<TTurnstile> P \<^bold>\<and> Q) \<longleftrightarrow> ((ex \<TTurnstile> P) \<and> (ex \<TTurnstile> Q))"
by (simp add: AND_def temp_sat_def satisfies_def)
lemma OR_temp_sat [simp]: "(ex \<TTurnstile> P \<^bold>\<or> Q) \<longleftrightarrow> ((ex \<TTurnstile> P) \<or> (ex \<TTurnstile> Q))"
by (simp add: OR_def temp_sat_def satisfies_def)
lemma NOT_temp_sat [simp]: "(ex \<TTurnstile> \<^bold>\<not> P) \<longleftrightarrow> (\<not> (ex \<TTurnstile> P))"
by (simp add: NOT_def temp_sat_def satisfies_def)
axiomatization
where mkfin_UU [simp]: "mkfin UU = nil"
and mkfin_nil [simp]: "mkfin nil = nil"
and mkfin_cons [simp]: "mkfin (a \<leadsto> s) = a \<leadsto> mkfin s"
lemmas [simp del] = HOL.ex_simps HOL.all_simps split_paired_Ex
setup \<open>map_theory_claset (fn ctxt => ctxt delSWrapper "split_all_tac")\<close>
subsection \<open>ex2seqC\<close>
lemma ex2seqC_unfold:
"ex2seqC =
(LAM ex. (\<lambda>s. case ex of
nil \<Rightarrow> (s, None, s) \<leadsto> nil
| x ## xs \<Rightarrow>
(flift1 (\<lambda>pr. (s, Some (fst pr), snd pr) \<leadsto> (ex2seqC $ xs) (snd pr)) $ x)))"
apply (rule trans)
apply (rule fix_eq4)
apply (rule ex2seqC_def)
apply (rule beta_cfun)
apply (simp add: flift1_def)
done
lemma ex2seqC_UU [simp]: "(ex2seqC $ UU) s = UU"
apply (subst ex2seqC_unfold)
apply simp
done
lemma ex2seqC_nil [simp]: "(ex2seqC $ nil) s = (s, None, s) \<leadsto> nil"
apply (subst ex2seqC_unfold)
apply simp
done
lemma ex2seqC_cons [simp]: "(ex2seqC $ ((a, t) \<leadsto> xs)) s = (s, Some a,t ) \<leadsto> (ex2seqC $ xs) t"
apply (rule trans)
apply (subst ex2seqC_unfold)
apply (simp add: Consq_def flift1_def)
apply (simp add: Consq_def flift1_def)
done
lemma ex2seq_UU: "ex2seq (s, UU) = (s, None, s) \<leadsto> nil"
by (simp add: ex2seq_def)
lemma ex2seq_nil: "ex2seq (s, nil) = (s, None, s) \<leadsto> nil"
by (simp add: ex2seq_def)
lemma ex2seq_cons: "ex2seq (s, (a, t) \<leadsto> ex) = (s, Some a, t) \<leadsto> ex2seq (t, ex)"
by (simp add: ex2seq_def)
declare ex2seqC_UU [simp del] ex2seqC_nil [simp del] ex2seqC_cons [simp del]
declare ex2seq_UU [simp] ex2seq_nil [simp] ex2seq_cons [simp]
lemma ex2seq_nUUnnil: "ex2seq exec \<noteq> UU \<and> ex2seq exec \<noteq> nil"
apply (tactic \<open>pair_tac @{context} "exec" 1\<close>)
apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
done
subsection \<open>Interface TL -- TLS\<close>
(* uses the fact that in executions states overlap, which is lost in
after the translation via ex2seq !! *)
lemma TL_TLS:
"\<forall>s a t. (P s) \<and> s \<midarrow>a\<midarrow>A\<rightarrow> t \<longrightarrow> (Q t)
\<Longrightarrow> ex \<TTurnstile> (Init (\<lambda>(s, a, t). P s) \<^bold>\<and> Init (\<lambda>(s, a, t). s \<midarrow>a\<midarrow>A\<rightarrow> t)
\<^bold>\<longrightarrow> (Next (Init (\<lambda>(s, a, t). Q s))))"
apply (unfold Init_def Next_def temp_sat_def satisfies_def IMPLIES_def AND_def)
apply clarify
apply (simp split add: split_if)
text \<open>\<open>TL = UU\<close>\<close>
apply (rule conjI)
apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
text \<open>\<open>TL = nil\<close>\<close>
apply (rule conjI)
apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
apply (tactic \<open>Seq_case_tac @{context} "x2" 1\<close>)
apply (simp add: unlift_def)
apply fast
apply (simp add: unlift_def)
apply fast
apply (simp add: unlift_def)
apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
text \<open>\<open>TL = cons\<close>\<close>
apply (simp add: unlift_def)
apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
done
end