(* Title: HOLCF/IOA/meta_theory/TLS.thy ID: $Id: TLS.thy,v 1.10 2006/05/28 17:54:22 wenzelm Exp $ Author: Olaf Müller *) header {* Temporal Logic of Steps -- tailored for I/O automata *} theory TLS imports IOA TL begin defaultsort type types ('a, 's) ioa_temp = "('a option,'s)transition temporal" ('a, 's) step_pred = "('a option,'s)transition predicate" 's state_pred = "'s predicate" consts option_lift :: "('a => 'b) => 'b => ('a option => 'b)" plift :: "('a => bool) => ('a option => bool)" temp_sat :: "('a,'s)execution => ('a,'s)ioa_temp => bool" (infixr "|==" 22) xt1 :: "'s predicate => ('a,'s)step_pred" xt2 :: "'a option predicate => ('a,'s)step_pred" validTE :: "('a,'s)ioa_temp => bool" validIOA :: "('a,'s)ioa => ('a,'s)ioa_temp => bool" mkfin :: "'a Seq => 'a Seq" ex2seq :: "('a,'s)execution => ('a option,'s)transition Seq" ex2seqC :: "('a,'s)pairs -> ('s => ('a option,'s)transition Seq)" defs mkfin_def: "mkfin s == if Partial s then @t. Finite t & s = t @@ UU else s" option_lift_def: "option_lift f s y == case y of None => s | Some x => (f x)" (* plift is used to determine that None action is always false in transition predicates *) plift_def: "plift P == option_lift P False" temp_sat_def: "ex |== P == ((ex2seq ex) |= P)" xt1_def: "xt1 P tr == P (fst tr)" xt2_def: "xt2 P tr == P (fst (snd tr))" ex2seq_def: "ex2seq ex == ((ex2seqC $(mkfin (snd ex))) (fst ex))" ex2seqC_def: "ex2seqC == (fix$(LAM h ex. (%s. case ex of nil => (s,None,s)>>nil | x##xs => (flift1 (%pr. (s,Some (fst pr), snd pr)>> (h$xs) (snd pr)) $x) )))" validTE_def: "validTE P == ! ex. (ex |== P)" validIOA_def: "validIOA A P == ! ex : executions A . (ex |== P)" axioms mkfin_UU: "mkfin UU = nil" mkfin_nil: "mkfin nil =nil" mkfin_cons: "(mkfin (a>>s)) = (a>>(mkfin s))" lemmas [simp del] = ex_simps all_simps split_paired_Ex declare Let_def [simp] ML_setup {* change_claset (fn cs => cs delSWrapper "split_all_tac") *} subsection {* ex2seqC *} lemma ex2seqC_unfold: "ex2seqC = (LAM ex. (%s. case ex of nil => (s,None,s)>>nil | x##xs => (flift1 (%pr. (s,Some (fst pr), snd pr)>> (ex2seqC$xs) (snd pr)) $x) ))" apply (rule trans) apply (rule fix_eq2) apply (rule ex2seqC_def) apply (rule beta_cfun) apply (simp add: flift1_def) done lemma ex2seqC_UU: "(ex2seqC $UU) s=UU" apply (subst ex2seqC_unfold) apply simp done lemma ex2seqC_nil: "(ex2seqC $nil) s = (s,None,s)>>nil" apply (subst ex2seqC_unfold) apply simp done lemma ex2seqC_cons: "(ex2seqC $((a,t)>>xs)) s = (s,Some a,t)>> ((ex2seqC$xs) t)" apply (rule trans) apply (subst ex2seqC_unfold) apply (simp add: Consq_def flift1_def) apply (simp add: Consq_def flift1_def) done declare ex2seqC_UU [simp] ex2seqC_nil [simp] ex2seqC_cons [simp] declare mkfin_UU [simp] mkfin_nil [simp] mkfin_cons [simp] lemma ex2seq_UU: "ex2seq (s, UU) = (s,None,s)>>nil" apply (simp add: ex2seq_def) done lemma ex2seq_nil: "ex2seq (s, nil) = (s,None,s)>>nil" apply (simp add: ex2seq_def) done lemma ex2seq_cons: "ex2seq (s, (a,t)>>ex) = (s,Some a,t) >> ex2seq (t, ex)" apply (simp add: ex2seq_def) done 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 ~= UU & ex2seq exec ~= nil" apply (tactic {* pair_tac "exec" 1 *}) apply (tactic {* Seq_case_simp_tac "y" 1 *}) apply (tactic {* pair_tac "a" 1 *}) done subsection {* Interface TL -- TLS *} (* uses the fact that in executions states overlap, which is lost in after the translation via ex2seq !! *) lemma TL_TLS: "[| ! s a t. (P s) & s-a--A-> t --> (Q t) |] ==> ex |== (Init (%(s,a,t). P s) .& Init (%(s,a,t). s -a--A-> t) .--> (Next (Init (%(s,a,t).Q s))))" apply (unfold Init_def Next_def temp_sat_def satisfies_def IMPLIES_def AND_def) apply (tactic "clarify_tac set_cs 1") apply (simp split add: split_if) (* TL = UU *) apply (rule conjI) apply (tactic {* pair_tac "ex" 1 *}) apply (tactic {* Seq_case_simp_tac "y" 1 *}) apply (tactic {* pair_tac "a" 1 *}) apply (tactic {* Seq_case_simp_tac "s" 1 *}) apply (tactic {* pair_tac "a" 1 *}) (* TL = nil *) apply (rule conjI) apply (tactic {* pair_tac "ex" 1 *}) apply (tactic {* Seq_case_tac "y" 1 *}) apply (simp add: unlift_def) apply fast apply (simp add: unlift_def) apply fast apply (simp add: unlift_def) apply (tactic {* pair_tac "a" 1 *}) apply (tactic {* Seq_case_simp_tac "s" 1 *}) apply (tactic {* pair_tac "a" 1 *}) (* TL =cons *) apply (simp add: unlift_def) apply (tactic {* pair_tac "ex" 1 *}) apply (tactic {* Seq_case_simp_tac "y" 1 *}) apply (tactic {* pair_tac "a" 1 *}) apply (tactic {* Seq_case_simp_tac "s" 1 *}) apply blast apply fastsimp apply (tactic {* pair_tac "a" 1 *}) apply fastsimp done end
lemma
(∃x. P x ∧ Q) = ((∃x. P x) ∧ Q)
(∃x. P ∧ Q x) = (P ∧ (∃x. Q x))
(∃x. P x ∨ Q) = ((∃x. P x) ∨ Q)
(∃x. P ∨ Q x) = (P ∨ (∃x. Q x))
(∃x. P x --> Q) = ((∀x. P x) --> Q)
(∃x. P --> Q x) = (P --> (∃x. Q x))
(∀x. P x ∧ Q) = ((∀x. P x) ∧ Q)
(∀x. P ∧ Q x) = (P ∧ (∀x. Q x))
(∀x. P x ∨ Q) = ((∀x. P x) ∨ Q)
(∀x. P ∨ Q x) = (P ∨ (∀x. Q x))
(∀x. P x --> Q) = ((∃x. P x) --> Q)
(∀x. P --> Q x) = (P --> (∀x. Q x))
(∃x. P x) = (∃a b. P (a, b))
lemma ex2seqC_unfold:
ex2seqC =
(LAM ex.
(λs. case ex of nil => [(s, None, s)!]
| x ## xs =>
(FLIFT pr. (s, Some (fst pr), snd pr)>>(ex2seqC·xs) (snd pr))·x))
lemma ex2seqC_UU:
(ex2seqC·UU) s = UU
lemma ex2seqC_nil:
(ex2seqC·nil) s = [(s, None, s)!]
lemma ex2seqC_cons:
(ex2seqC·((a, t)>>xs)) s = (s, Some a, t)>>(ex2seqC·xs) t
lemma ex2seq_UU:
ex2seq (s, UU) = [(s, None, s)!]
lemma ex2seq_nil:
ex2seq (s, nil) = [(s, None, s)!]
lemma ex2seq_cons:
ex2seq (s, (a, t)>>ex) = (s, Some a, t)>>ex2seq (t, ex)
lemma ex2seq_nUUnnil:
ex2seq exec ≠ UU ∧ ex2seq exec ≠ nil
lemma TL_TLS:
∀s a t. P s ∧ s -a--A-> t --> Q t
==> ex |==
<λ(s, a, t). P s> ∧ <λ(s, a, t). s -a--A-> t> --> Next <λ(s, a, t). Q s>