diff -r 17305a85493d -r c495eb16beb6 CpsG.thy_1_1
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/CpsG.thy_1_1	Wed Jan 27 19:28:42 2016 +0800
@@ -0,0 +1,1751 @@
+theory CpsG
+imports PIPDefs 
+begin
+
+lemma Max_f_mono:
+  assumes seq: "A \<subseteq> B"
+  and np: "A \<noteq> {}"
+  and fnt: "finite B"
+  shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+  from seq show "f ` A \<subseteq> f ` B" by auto
+next
+  from np show "f ` A \<noteq> {}" by auto
+next
+  from fnt and seq show "finite (f ` B)" by auto
+qed
+
+
+locale valid_trace = 
+  fixes s
+  assumes vt : "vt s"
+
+locale valid_trace_e = valid_trace +
+  fixes e
+  assumes vt_e: "vt (e#s)"
+begin
+
+lemma pip_e: "PIP s e"
+  using vt_e by (cases, simp)  
+
+end
+
+locale valid_trace_create = valid_trace_e + 
+  fixes th prio
+  assumes is_create: "e = Create th prio"
+
+locale valid_trace_exit = valid_trace_e + 
+  fixes th
+  assumes is_exit: "e = Exit th"
+
+locale valid_trace_p = valid_trace_e + 
+  fixes th cs
+  assumes is_p: "e = P th cs"
+
+locale valid_trace_v = valid_trace_e + 
+  fixes th cs
+  assumes is_v: "e = V th cs"
+begin
+  definition "rest = tl (wq s cs)"
+  definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+end
+
+locale valid_trace_v_n = valid_trace_v +
+  assumes rest_nnl: "rest \<noteq> []"
+
+locale valid_trace_v_e = valid_trace_v +
+  assumes rest_nil: "rest = []"
+
+locale valid_trace_set= valid_trace_e + 
+  fixes th prio
+  assumes is_set: "e = Set th prio"
+
+context valid_trace
+begin
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes "PP []"
+     and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
+                   PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
+     shows "PP s"
+proof(induct rule:vt.induct[OF vt, case_names Init Step])
+  case Init
+  from assms(1) show ?case .
+next
+  case (Step s e)
+  show ?case
+  proof(rule assms(2))
+    show "valid_trace_e s e" using Step by (unfold_locales, auto)
+  next
+    show "PP s" using Step by simp
+  next
+    show "PIP s e" using Step by simp
+  qed
+qed
+
+lemma  vt_moment: "\<And> t. vt (moment t s)"
+proof(induct rule:ind)
+  case Nil
+  thus ?case by (simp add:vt_nil)
+next
+  case (Cons s e t)
+  show ?case
+  proof(cases "t \<ge> length (e#s)")
+    case True
+    from True have "moment t (e#s) = e#s" by simp
+    thus ?thesis using Cons
+      by (simp add:valid_trace_def valid_trace_e_def, auto)
+  next
+    case False
+    from Cons have "vt (moment t s)" by simp
+    moreover have "moment t (e#s) = moment t s"
+    proof -
+      from False have "t \<le> length s" by simp
+      from moment_app [OF this, of "[e]"] 
+      show ?thesis by simp
+    qed
+    ultimately show ?thesis by simp
+  qed
+qed
+
+lemma finite_threads:
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+apply(simp add: Let_def cpreced_initial)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, auto)
+
+locale valid_moment = valid_trace + 
+  fixes i :: nat
+
+sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
+  by (unfold_locales, insert vt_moment, auto)
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+  by  (unfold s_waiting_def cs_waiting_def wq_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def wq_def cs_holding_def, simp)
+
+lemma runing_ready: 
+  shows "runing s \<subseteq> readys s"
+  unfolding runing_def readys_def
+  by auto 
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+  unfolding readys_def
+  by auto
+
+lemma wq_v_neq [simp]:
+   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
+  by (auto simp:wq_def Let_def cp_def split:list.splits)
+
+lemma runing_head:
+  assumes "th \<in> runing s"
+  and "th \<in> set (wq_fun (schs s) cs)"
+  shows "th = hd (wq_fun (schs s) cs)"
+  using assms
+  by (simp add:runing_def readys_def s_waiting_def wq_def)
+
+context valid_trace
+begin
+
+lemma runing_wqE:
+  assumes "th \<in> runing s"
+  and "th \<in> set (wq s cs)"
+  obtains rest where "wq s cs = th#rest"
+proof -
+  from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
+    by (meson list.set_cases)
+  have "th' = th"
+  proof(rule ccontr)
+    assume "th' \<noteq> th"
+    hence "th \<noteq> hd (wq s cs)" using eq_wq by auto 
+    with assms(2)
+    have "waiting s th cs" 
+      by (unfold s_waiting_def, fold wq_def, auto)
+    with assms show False 
+      by (unfold runing_def readys_def, auto)
+  qed
+  with eq_wq that show ?thesis by metis
+qed
+
+end
+
+context valid_trace_create
+begin
+
+lemma wq_neq_simp [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_create wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace_exit
+begin
+
+lemma wq_neq_simp [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_exit wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace_p
+begin
+
+lemma wq_neq_simp [simp]:
+  assumes "cs' \<noteq> cs"
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_p wq_def
+  by (auto simp:Let_def)
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_p]
+  show ?thesis by (cases, simp)
+qed
+
+lemma th_not_waiting: 
+  "\<not> waiting s th c"
+proof -
+  have "th \<in> readys s"
+    using runing_ready runing_th_s by blast 
+  thus ?thesis
+    by (unfold readys_def, auto)
+qed
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+lemma th_not_in_wq: 
+  shows "th \<notin> set (wq s cs)"
+proof
+  assume otherwise: "th \<in> set (wq s cs)"
+  from runing_wqE[OF runing_th_s this]
+  obtain rest where eq_wq: "wq s cs = th#rest" by blast
+  with otherwise
+  have "holding s th cs"
+    by (unfold s_holding_def, fold wq_def, simp)
+  hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
+    by (unfold s_RAG_def, fold holding_eq, auto)
+  from pip_e[unfolded is_p]
+  show False
+  proof(cases)
+    case (thread_P)
+    with cs_th_RAG show ?thesis by auto
+  qed
+qed
+
+lemma wq_es_cs: 
+  "wq (e#s) cs =  wq s cs @ [th]"
+  by (unfold is_p wq_def, auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+proof(cases "cs' = cs")
+  case True
+  show ?thesis using True assms th_not_in_wq
+    by (unfold True wq_es_cs, auto)
+qed (insert assms, simp)
+
+end
+
+context valid_trace_v
+begin
+
+lemma wq_neq_simp [simp]:
+  assumes "cs' \<noteq> cs"
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_v wq_def
+  by (auto simp:Let_def)
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis by (cases, simp)
+qed
+
+lemma th_not_waiting: 
+  "\<not> waiting s th c"
+proof -
+  have "th \<in> readys s"
+    using runing_ready runing_th_s by blast 
+  thus ?thesis
+    by (unfold readys_def, auto)
+qed
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+lemma wq_s_cs:
+  "wq s cs = th#rest"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis
+  proof(cases)
+    case (thread_V)
+    from this(2) show ?thesis
+      by (unfold rest_def s_holding_def, fold wq_def,
+                 metis empty_iff list.collapse list.set(1))
+  qed
+qed
+
+lemma wq_es_cs:
+  "wq (e#s) cs = wq'"
+ using wq_s_cs[unfolded wq_def]
+ by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) 
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+proof(cases "cs' = cs")
+  case True
+  show ?thesis
+  proof(unfold True wq_es_cs wq'_def, rule someI2)
+    show "distinct rest \<and> set rest = set rest"
+        using assms[unfolded True wq_s_cs] by auto
+  qed simp
+qed (insert assms, simp)
+
+end
+
+context valid_trace_set
+begin
+
+lemma wq_neq_simp [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_set wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace
+begin
+
+lemma actor_inv: 
+  assumes "PIP s e"
+  and "\<not> isCreate e"
+  shows "actor e \<in> runing s"
+  using assms
+  by (induct, auto)
+
+lemma isP_E:
+  assumes "isP e"
+  obtains cs where "e = P (actor e) cs"
+  using assms by (cases e, auto)
+
+lemma isV_E:
+  assumes "isV e"
+  obtains cs where "e = V (actor e) cs"
+  using assms by (cases e, auto) 
+
+lemma wq_distinct: "distinct (wq s cs)"
+proof(induct rule:ind)
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case 
+  proof(cases e)
+    case (Create th prio)
+    interpret vt_create: valid_trace_create s e th prio 
+      using Create by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept) 
+  next
+    case (Exit th)
+    interpret vt_exit: valid_trace_exit s e th  
+        using Exit by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept) 
+  next
+    case (P th cs)
+    interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) 
+  next
+    case (V th cs)
+    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) 
+  next
+    case (Set th prio)
+    interpret vt_set: valid_trace_set s e th prio
+        using Set by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) 
+  qed
+qed (unfold wq_def Let_def, simp)
+
+end
+
+context valid_trace_e
+begin
+
+text {*
+  The following lemma shows that only the @{text "P"}
+  operation can add new thread into waiting queues. 
+  Such kind of lemmas are very obvious, but need to be checked formally.
+  This is a kind of confirmation that our modelling is correct.
+*}
+
+lemma wq_in_inv: 
+  assumes s_ni: "thread \<notin> set (wq s cs)"
+  and s_i: "thread \<in> set (wq (e#s) cs)"
+  shows "e = P thread cs"
+proof(cases e)
+  -- {* This is the only non-trivial case: *}
+  case (V th cs1)
+  have False
+  proof(cases "cs1 = cs")
+    case True
+    show ?thesis
+    proof(cases "(wq s cs1)")
+      case (Cons w_hd w_tl)
+      have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
+      proof -
+        have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
+          using  Cons V by (auto simp:wq_def Let_def True split:if_splits)
+        moreover have "set ... \<subseteq> set (wq s cs)"
+        proof(rule someI2)
+          show "distinct w_tl \<and> set w_tl = set w_tl"
+            by (metis distinct.simps(2) local.Cons wq_distinct)
+        qed (insert Cons True, auto)
+        ultimately show ?thesis by simp
+      qed
+      with assms show ?thesis by auto
+    qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
+  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+  thus ?thesis by auto
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+
+lemma wq_out_inv: 
+  assumes s_in: "thread \<in> set (wq s cs)"
+  and s_hd: "thread = hd (wq s cs)"
+  and s_i: "thread \<noteq> hd (wq (e#s) cs)"
+  shows "e = V thread cs"
+proof(cases e)
+-- {* There are only two non-trivial cases: *}
+  case (V th cs1)
+  show ?thesis
+  proof(cases "cs1 = cs")
+    case True
+    have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
+    thus ?thesis
+    proof(cases)
+      case (thread_V)
+      moreover have "th = thread" using thread_V(2) s_hd
+          by (unfold s_holding_def wq_def, simp)
+      ultimately show ?thesis using V True by simp
+    qed
+  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+next
+  case (P th cs1)
+  show ?thesis
+  proof(cases "cs1 = cs")
+    case True
+    with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
+      by (auto simp:wq_def Let_def split:if_splits)
+    with s_i s_hd s_in have False
+      by (metis empty_iff hd_append2 list.set(1) wq_def) 
+    thus ?thesis by simp
+  qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+
+end
+
+
+context valid_trace
+begin
+
+
+text {* (* ddd *)
+  The nature of the work is like this: since it starts from a very simple and basic 
+  model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
+  For instance, the fact 
+  that one thread can not be blocked by two critical resources at the same time
+  is obvious, because only running threads can make new requests, if one is waiting for 
+  a critical resource and get blocked, it can not make another resource request and get 
+  blocked the second time (because it is not running). 
+
+  To derive this fact, one needs to prove by contraction and 
+  reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
+  named @{text "p_split"}, which is about status changing along the time axis. It says if 
+  a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
+  but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} 
+  in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history 
+  of events leading to it), such that @{text "Q"} switched 
+  from being @{text "False"} to @{text "True"} and kept being @{text "True"}
+  till the last moment of @{text "s"}.
+
+  Suppose a thread @{text "th"} is blocked
+  on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, 
+  since no thread is blocked at the very beginning, by applying 
+  @{text "p_split"} to these two blocking facts, there exist 
+  two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that 
+  @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} 
+  and kept on blocked on them respectively ever since.
+ 
+  Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
+  However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
+  in blocked state at moment @{text "t2"} and could not
+  make any request and get blocked the second time: Contradiction.
+*}
+
+lemma waiting_unique_pre: (* ddd *)
+  assumes h11: "thread \<in> set (wq s cs1)"
+  and h12: "thread \<noteq> hd (wq s cs1)"
+  assumes h21: "thread \<in> set (wq s cs2)"
+  and h22: "thread \<noteq> hd (wq s cs2)"
+  and neq12: "cs1 \<noteq> cs2"
+  shows "False"
+proof -
+  let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  from h11 and h12 have q1: "?Q cs1 s" by simp
+  from h21 and h22 have q2: "?Q cs2 s" by simp
+  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+  from p_split [of "?Q cs1", OF q1 nq1]
+  obtain t1 where lt1: "t1 < length s"
+    and np1: "\<not> ?Q cs1 (moment t1 s)"
+    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
+  from p_split [of "?Q cs2", OF q2 nq2]
+  obtain t2 where lt2: "t2 < length s"
+    and np2: "\<not> ?Q cs2 (moment t2 s)"
+    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
+  { fix s cs
+    assume q: "?Q cs s"
+    have "thread \<notin> runing s"
+    proof
+      assume "thread \<in> runing s"
+      hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and> 
+                 thread \<noteq> hd (wq_fun (schs s) cs))"
+        by (unfold runing_def s_waiting_def readys_def, auto)
+      from this[rule_format, of cs] q 
+      show False by (simp add: wq_def) 
+    qed
+  } note q_not_runing = this
+  { fix t1 t2 cs1 cs2
+    assume  lt1: "t1 < length s"
+    and np1: "\<not> ?Q cs1 (moment t1 s)"
+    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
+    and lt2: "t2 < length s"
+    and np2: "\<not> ?Q cs2 (moment t2 s)"
+    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
+    and lt12: "t1 < t2"
+    let ?t3 = "Suc t2"
+    from lt2 have le_t3: "?t3 \<le> length s" by auto
+    from moment_plus [OF this] 
+    obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+    have "t2 < ?t3" by simp
+    from nn2 [rule_format, OF this] and eq_m
+    have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+         h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+    have "vt (e#moment t2 s)"
+    proof -
+      from vt_moment 
+      have "vt (moment ?t3 s)" .
+      with eq_m show ?thesis by simp
+    qed
+    then interpret vt_e: valid_trace_e "moment t2 s" "e"
+        by (unfold_locales, auto, cases, simp)
+    have ?thesis
+    proof -
+      have "thread \<in> runing (moment t2 s)"
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        have "e = V thread cs2"
+        proof -
+          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
+              using True and np2  by auto 
+          from vt_e.wq_out_inv[OF True this h2]
+          show ?thesis .
+        qed
+        thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto
+      next
+        case False
+        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+        with vt_e.actor_inv[OF vt_e.pip_e]
+        show ?thesis by auto
+      qed
+      moreover have "thread \<notin> runing (moment t2 s)"
+        by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
+      ultimately show ?thesis by simp
+    qed
+  } note lt_case = this
+  show ?thesis
+  proof -
+    { assume "t1 < t2"
+      from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
+      have ?thesis .
+    } moreover {
+      assume "t2 < t1"
+      from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
+      have ?thesis .
+    } moreover {
+      assume eq_12: "t1 = t2"
+      let ?t3 = "Suc t2"
+      from lt2 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+      have lt_2: "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+           h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+      from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12]
+      have g1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+           g2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have "vt (e#moment t2 s)"
+      proof -
+        from vt_moment 
+        have "vt (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      then interpret vt_e: valid_trace_e "moment t2 s" "e"
+          by (unfold_locales, auto, cases, simp)
+      have "e = V thread cs2 \<or> e = P thread cs2"
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        have "e = V thread cs2"
+        proof -
+          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
+              using True and np2  by auto 
+          from vt_e.wq_out_inv[OF True this h2]
+          show ?thesis .
+        qed
+        thus ?thesis by auto
+      next
+        case False
+        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+        thus ?thesis by auto
+      qed
+      moreover have "e = V thread cs1 \<or> e = P thread cs1"
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        have eq_th: "thread = hd (wq (moment t1 s) cs1)" 
+              using True and np1  by auto 
+        from vt_e.wq_out_inv[folded eq_12, OF True this g2]
+        have "e = V thread cs1" .
+        thus ?thesis by auto
+      next
+        case False
+        have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
+        thus ?thesis by auto
+      qed
+      ultimately have ?thesis using neq12 by auto
+    } ultimately show ?thesis using nat_neq_iff by blast 
+  qed
+qed
+
+text {*
+  This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
+*}
+
+lemma waiting_unique:
+  assumes "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+  using waiting_unique_pre assms
+  unfolding wq_def s_waiting_def
+  by auto
+
+end
+
+(* not used *)
+text {*
+  Every thread can only be blocked on one critical resource, 
+  symmetrically, every critical resource can only be held by one thread. 
+  This fact is much more easier according to our definition. 
+*}
+lemma held_unique:
+  assumes "holding (s::event list) th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+ by (insert assms, unfold s_holding_def, auto)
+
+lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits)
+
+lemma last_set_unique: 
+  "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
+          \<Longrightarrow> th1 = th2"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits dest:last_set_lt)
+
+lemma preced_unique : 
+  assumes pcd_eq: "preced th1 s = preced th2 s"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "th1 = th2"
+proof -
+  from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
+  from last_set_unique [OF this th_in1 th_in2]
+  show ?thesis .
+qed
+                      
+lemma preced_linorder: 
+  assumes neq_12: "th1 \<noteq> th2"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
+proof -
+  from preced_unique [OF _ th_in1 th_in2] and neq_12 
+  have "preced th1 s \<noteq> preced th2 s" by auto
+  thus ?thesis by auto
+qed
+
+text {*
+  The following three lemmas show that @{text "RAG"} does not change
+  by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
+  events, respectively.
+*}
+
+lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+
+context valid_trace_v
+begin
+
+lemma distinct_rest: "distinct rest"
+  by (simp add: distinct_tl rest_def wq_distinct)
+
+lemma holding_cs_eq_th:
+  assumes "holding s t cs"
+  shows "t = th"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis
+  proof(cases)
+    case (thread_V)
+    from held_unique[OF this(2) assms]
+    show ?thesis by simp
+  qed
+qed
+
+lemma distinct_wq': "distinct wq'"
+  by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
+  
+lemma th'_in_inv:
+  assumes "th' \<in> set wq'"
+  shows "th' \<in> set rest"
+  using assms
+  by (metis (mono_tags, lifting) distinct.simps(2) 
+        rest_def some_eq_ex wq'_def wq_distinct wq_s_cs) 
+
+lemma neq_t_th: 
+  assumes "waiting (e#s) t c"
+  shows "t \<noteq> th"
+proof
+  assume otherwise: "t = th"
+  show False
+  proof(cases "c = cs")
+    case True
+    have "t \<in> set wq'" 
+     using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
+     by simp 
+    from th'_in_inv[OF this] have "t \<in> set rest" .
+    with wq_s_cs[folded otherwise] wq_distinct[of cs]
+    show ?thesis by simp
+  next
+    case False
+    have "wq (e#s) c = wq s c" using False
+        by (unfold is_v, simp)
+    hence "waiting s t c" using assms 
+        by (simp add: cs_waiting_def waiting_eq)
+    hence "t \<notin> readys s" by (unfold readys_def, auto)
+    hence "t \<notin> runing s" using runing_ready by auto 
+    with runing_th_s[folded otherwise] show ?thesis by auto
+  qed
+qed
+
+lemma waiting_esI1:
+  assumes "waiting s t c"
+      and "c \<noteq> cs" 
+  shows "waiting (e#s) t c" 
+proof -
+  have "wq (e#s) c = wq s c" 
+    using assms(2) is_v by auto
+  with assms(1) show ?thesis 
+    using cs_waiting_def waiting_eq by auto 
+qed
+
+lemma holding_esI2:
+  assumes "c \<noteq> cs" 
+  and "holding s t c"
+  shows "holding (e#s) t c"
+proof -
+  from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
+  from assms(2)[unfolded s_holding_def, folded wq_def, 
+                folded this, unfolded wq_def, folded s_holding_def]
+  show ?thesis .
+qed
+
+lemma holding_esI1:
+  assumes "holding s t c"
+  and "t \<noteq> th"
+  shows "holding (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms using holding_cs_eq_th by blast 
+  from holding_esI2[OF this assms(1)]
+  show ?thesis .
+qed
+
+end
+
+context valid_trace_v_n
+begin
+
+lemma neq_wq': "wq' \<noteq> []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x \<noteq> []" using rest_nnl by auto
+qed 
+
+definition "taker = hd wq'"
+
+definition "rest' = tl wq'"
+
+lemma eq_wq': "wq' = taker # rest'"
+  by (simp add: neq_wq' rest'_def taker_def)
+
+lemma next_th_taker: 
+  shows "next_th s th cs taker"
+  using rest_nnl taker_def wq'_def wq_s_cs 
+  by (auto simp:next_th_def)
+
+lemma taker_unique: 
+  assumes "next_th s th cs taker'"
+  shows "taker' = taker"
+proof -
+  from assms
+  obtain rest' where 
+    h: "wq s cs = th # rest'" 
+       "taker' = hd (SOME q. distinct q \<and> set q = set rest')"
+          by (unfold next_th_def, auto)
+  with wq_s_cs have "rest' = rest" by auto
+  thus ?thesis using h(2) taker_def wq'_def by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
+  by (smt all_not_in_conv bot.extremum insertI1 insert_subset 
+      mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {(Cs cs, Th taker)}"
+  using next_th_taker taker_def waiting_set_eq 
+  by fastforce
+   
+lemma holding_taker:
+  shows "holding (e#s) taker cs"
+    by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, 
+        auto simp:neq_wq' taker_def)
+
+lemma waiting_esI2:
+  assumes "waiting s t cs"
+      and "t \<noteq> taker"
+  shows "waiting (e#s) t cs" 
+proof -
+  have "t \<in> set wq'" 
+  proof(unfold wq'_def, rule someI2)
+    show "distinct rest \<and> set rest = set rest"
+          by (simp add: distinct_rest)
+  next
+    fix x
+    assume "distinct x \<and> set x = set rest"
+    moreover have "t \<in> set rest"
+        using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto 
+    ultimately show "t \<in> set x" by simp
+  qed
+  moreover have "t \<noteq> hd wq'"
+    using assms(2) taker_def by auto 
+  ultimately show ?thesis
+    by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+     |    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
+  have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
+  hence "t \<noteq> taker" by (simp add: taker_def) 
+  moreover hence "t \<noteq> th" using assms neq_t_th by blast 
+  moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv) 
+  ultimately have "waiting s t cs"
+    by (metis cs_waiting_def list.distinct(2) list.sel(1) 
+                list.set_sel(2) rest_def waiting_eq wq_s_cs)  
+  show ?thesis using that(2)
+  using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto   
+qed
+
+lemma holding_esI1:
+  assumes "c = cs"
+  and "t = taker"
+  shows "holding (e#s) t c"
+  by (unfold assms, simp add: holding_taker)
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c = cs" "t = taker"
+      | "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from assms[unfolded True, unfolded s_holding_def, 
+             folded wq_def, unfolded wq_es_cs]
+  have "t = taker" by (simp add: taker_def) 
+  from that(1)[OF True this] show ?thesis .
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that(2)[OF False this] show ?thesis .
+qed
+
+end 
+
+
+context valid_trace_v_e
+begin
+
+lemma nil_wq': "wq' = []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x = []" using rest_nil by auto
+qed 
+
+lemma no_taker: 
+  assumes "next_th s th cs taker"
+  shows "False"
+proof -
+  from assms[unfolded next_th_def]
+  obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
+    by auto
+  thus ?thesis using rest_def rest_nil by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
+  using no_taker by auto
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
+  using no_taker by auto
+   
+lemma no_holding:
+  assumes "holding (e#s) taker cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_holding_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma no_waiting:
+  assumes "waiting (e#s) t cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma waiting_esI2:
+  assumes "waiting s t c"
+  shows "waiting (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms
+    using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto 
+  from waiting_esI1[OF assms this]
+  show ?thesis .
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from no_waiting[OF assms[unfolded True]]
+  show ?thesis by auto
+qed
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from no_holding[OF assms[unfolded True]] 
+  show ?thesis by auto
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that[OF False this] show ?thesis .
+qed
+
+end 
+
+lemma rel_eqI:
+  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
+  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
+  shows "A = B"
+  using assms by auto
+
+lemma in_RAG_E:
+  assumes "(n1, n2) \<in> RAG (s::state)"
+  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
+      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
+  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
+  by auto
+  
+context valid_trace_v
+begin
+
+lemma RAG_es:
+  "RAG (e # s) =
+   RAG s - {(Cs cs, Th th)} -
+     {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+     {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R"
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_n.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      next
+        case 2
+        with waiting(1,2)
+        show ?thesis
+         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      qed
+    next
+      case True
+      interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_e.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      qed
+    qed
+  next
+    case (holding th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from holding(3)
+      show ?thesis
+      proof(cases rule:h_n.holding_esE)
+        case 1
+        with holding(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      next
+        case 2
+        with holding(1,2)
+        show ?thesis
+         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold holding_eq, auto)
+      qed
+    next
+      case True
+      interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+      from holding(3)
+      show ?thesis
+      proof(cases rule:h_e.holding_esE)
+        case 1
+        with holding(1,2)
+        show ?thesis
+        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
+             fold holding_eq, auto)
+      qed
+    qed
+  qed
+next
+  fix n1 n2
+  assume h: "(n1, n2) \<in> ?R"
+  show "(n1, n2) \<in> ?L"
+  proof(cases "rest = []")
+    case False
+    interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+    from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
+    have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
+                            \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or> 
+          (n2 = Th h_n.taker \<and> n1 = Cs cs)" 
+      by auto
+   thus ?thesis
+   proof
+      assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
+      with h_n.holding_taker
+      show ?thesis 
+        by (unfold s_RAG_def, fold holding_eq, auto)
+   next
+    assume h: "(n1, n2) \<in> RAG s \<and>
+        (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
+    hence "(n1, n2) \<in> RAG s" by simp
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from h and this(1,2)
+      have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
+      hence "waiting (e#s) th' cs'" 
+      proof
+        assume "cs' \<noteq> cs"
+        from waiting_esI1[OF waiting(3) this] 
+        show ?thesis .
+      next
+        assume neq_th': "th' \<noteq> h_n.taker"
+        show ?thesis
+        proof(cases "cs' = cs")
+          case False
+          from waiting_esI1[OF waiting(3) this] 
+          show ?thesis .
+        next
+          case True
+          from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
+          show ?thesis .
+        qed
+      qed
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from h this(1,2)
+      have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+      hence "holding (e#s) th' cs'"
+      proof
+        assume "cs' \<noteq> cs"
+        from holding_esI2[OF this holding(3)] 
+        show ?thesis .
+      next
+        assume "th' \<noteq> th"
+        from holding_esI1[OF holding(3) this]
+        show ?thesis .
+      qed
+      thus ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+   qed
+ next
+   case True
+   interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+   from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
+   have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)" 
+      by auto
+   from h_s(1)
+   show ?thesis
+   proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from h_e.waiting_esI2[OF this(3)]
+    show ?thesis using waiting(1,2)
+      by (unfold s_RAG_def, fold waiting_eq, auto)
+   next
+    case (holding th' cs')
+    with h_s(2)
+    have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+    thus ?thesis
+    proof
+      assume neq_cs: "cs' \<noteq> cs"
+      from holding_esI2[OF this holding(3)]
+      show ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    next
+      assume "th' \<noteq> th"
+      from holding_esI1[OF holding(3) this]
+      show ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+   qed
+ qed
+qed
+
+end
+
+lemma step_RAG_v: 
+assumes vt:
+  "vt (V th cs#s)"
+shows "
+  RAG (V th cs # s) =
+  RAG s - {(Cs cs, Th th)} -
+  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+  {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof -
+  interpret vt_v: valid_trace_v s "V th cs"
+    using assms step_back_vt by (unfold_locales, auto) 
+  show ?thesis using vt_v.RAG_es .
+qed
+
+lemma (in valid_trace_create)
+  th_not_in_threads: "th \<notin> threads s"
+proof -
+  from pip_e[unfolded is_create]
+  show ?thesis by (cases, simp)
+qed
+
+lemma (in valid_trace_create)
+  threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
+  by (unfold is_create, simp)
+
+lemma (in valid_trace_exit)
+  threads_es [simp]: "threads (e#s) = threads s - {th}"
+  by (unfold is_exit, simp)
+
+lemma (in valid_trace_p)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_p, simp)
+
+lemma (in valid_trace_v)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_v, simp)
+
+lemma (in valid_trace_v)
+  th_not_in_rest[simp]: "th \<notin> set rest"
+proof
+  assume otherwise: "th \<in> set rest"
+  have "distinct (wq s cs)" by (simp add: wq_distinct)
+  from this[unfolded wq_s_cs] and otherwise
+  show False by auto
+qed
+
+lemma (in valid_trace_v)
+  set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
+proof(unfold wq_es_cs wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest)
+next
+  fix x
+  assume "distinct x \<and> set x = set rest"
+  thus "set x = set (wq s cs) - {th}" 
+      by (unfold wq_s_cs, simp)
+qed
+
+lemma (in valid_trace_exit)
+  th_not_in_wq: "th \<notin> set (wq s cs)"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis
+  by (cases, unfold holdents_def s_holding_def, fold wq_def, 
+             auto elim!:runing_wqE)
+qed
+
+lemma (in valid_trace) wq_threads: 
+  assumes "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+  using assms
+proof(induct rule:ind)
+  case (Nil)
+  thus ?case by (auto simp:wq_def)
+next
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case
+  proof(cases e)
+    case (Create th' prio')
+    interpret vt: valid_trace_create s e th' prio'
+      using Create by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems by auto
+  next
+    case (Exit th')
+    interpret vt: valid_trace_exit s e th'
+      using Exit by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto 
+  next
+    case (P th' cs')
+    interpret vt: valid_trace_p s e th' cs'
+      using P by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems readys_threads 
+        runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv 
+        by fastforce 
+  next
+    case (V th' cs')
+    interpret vt: valid_trace_v s e th' cs'
+      using V by (unfold_locales, simp)
+    show ?thesis using Cons
+      using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
+  next
+    case (Set th' prio)
+    interpret vt: valid_trace_set s e th' prio
+      using Set by (unfold_locales, simp)
+    show ?thesis using Cons.hyps(2) Cons.prems vt.is_set 
+        by (auto simp:wq_def Let_def)
+  qed
+qed 
+
+context valid_trace
+begin
+
+lemma  dm_RAG_threads:
+  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
+  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
+  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
+  hence "th \<in> set (wq s cs)"
+    by (unfold s_RAG_def, auto simp:cs_waiting_def)
+  from wq_threads [OF this] show ?thesis .
+qed
+
+lemma  cp_le:
+  assumes th_in: "th \<in> threads s"
+  shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
+  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
+         \<le> Max ((\<lambda>th. preced th s) ` threads s)"
+    (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
+  proof(rule Max_f_mono)
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
+  next
+    from finite_threads
+    show "finite (threads s)" .
+  next
+    from th_in
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
+      apply (auto simp:Domain_def)
+      apply (rule_tac dm_RAG_threads)
+      apply (unfold trancl_domain [of "RAG s", symmetric])
+      by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
+  qed
+qed
+
+lemma max_cp_eq: 
+  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
+  (is "?l = ?r")
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  have "?l \<in> ((cp s) ` threads s)"
+  proof(rule Max_in)
+    from finite_threads
+    show "finite (cp s ` threads s)" by auto
+  next
+    from False show "cp s ` threads s \<noteq> {}" by auto
+  qed
+  then obtain th 
+    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
+  have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
+  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
+  proof -
+    have "?r \<in> (?f ` ?A)"
+    proof(rule Max_in)
+      from finite_threads
+      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
+    next
+      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
+    qed
+    then obtain th' where 
+      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
+    from le_cp [of th']  eq_r
+    have "?r \<le> cp s th'" 
+    moreover have "\<dots> \<le> cp s th"
+    proof(fold eq_l)
+      show " cp s th' \<le> Max (cp s ` threads s)"
+      proof(rule Max_ge)
+        from th_in' show "cp s th' \<in> cp s ` threads s"
+          by auto
+      next
+        from finite_threads
+        show "finite (cp s ` threads s)" by auto
+      qed
+    qed
+    ultimately show ?thesis by auto
+  qed
+  ultimately show ?thesis using eq_l by auto
+qed
+
+lemma max_cp_eq_the_preced:
+  shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
+  using max_cp_eq using the_preced_def by presburger 
+
+end
+
+lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
+  by (unfold preced_def, simp)
+
+lemma (in valid_trace_v)
+  preced_es: "preced th (e#s) = preced th s"
+  by (unfold is_v preced_def, simp)
+
+lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
+proof
+  fix th'
+  show "the_preced (V th cs # s) th' = the_preced s th'"
+    by (unfold the_preced_def preced_def, simp)
+qed
+
+lemma (in valid_trace_v)
+  the_preced_es: "the_preced (e#s) = the_preced s"
+  by (unfold is_v preced_def, simp)
+
+context valid_trace_p
+begin
+
+lemma not_holding_es_th_cs: "\<not> holding s th cs"
+proof
+  assume otherwise: "holding s th cs"
+  from pip_e[unfolded is_p]
+  show False
+  proof(cases)
+    case (thread_P)
+    moreover have "(Cs cs, Th th) \<in> RAG s"
+      using otherwise cs_holding_def 
+            holding_eq th_not_in_wq by auto
+    ultimately show ?thesis by auto
+  qed
+qed
+
+lemma waiting_kept:
+  assumes "waiting s th' cs'"
+  shows "waiting (e#s) th' cs'"
+  using assms
+  by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2) 
+      rotate1.simps(2) self_append_conv2 set_rotate1 
+        th_not_in_wq waiting_eq wq_es_cs wq_neq_simp)
+  
+lemma holding_kept:
+  assumes "holding s th' cs'"
+  shows "holding (e#s) th' cs'"
+proof(cases "cs' = cs")
+  case False
+  hence "wq (e#s) cs' = wq s cs'" by simp
+  with assms show ?thesis using cs_holding_def holding_eq by auto 
+next
+  case True
+  from assms[unfolded s_holding_def, folded wq_def]
+  obtain rest where eq_wq: "wq s cs' = th'#rest"
+    by (metis empty_iff list.collapse list.set(1)) 
+  hence "wq (e#s) cs' = th'#(rest@[th])"
+    by (simp add: True wq_es_cs) 
+  thus ?thesis
+    by (simp add: cs_holding_def holding_eq) 
+qed
+
+end
+
+locale valid_trace_p_h = valid_trace_p +
+  assumes we: "wq s cs = []"
+
+locale valid_trace_p_w = valid_trace_p +
+  assumes wne: "wq s cs \<noteq> []"
+begin
+
+definition "holder = hd (wq s cs)"
+definition "waiters = tl (wq s cs)"
+definition "waiters' = waiters @ [th]"
+
+lemma wq_s_cs: "wq s cs = holder#waiters"
+    by (simp add: holder_def waiters_def wne)
+    
+lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
+  by (simp add: wq_es_cs wq_s_cs)
+
+lemma waiting_es_th_cs: "waiting (e#s) th cs"
+  using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
+
+lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
+   by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
+
+lemma holding_esE:
+  assumes "holding (e#s) th' cs'"
+  obtains "holding s th' cs'"
+  using assms 
+proof(cases "cs' = cs")
+  case False
+  hence "wq (e#s) cs' = wq s cs'" by simp
+  with assms show ?thesis
+    using cs_holding_def holding_eq that by auto 
+next
+  case True
+  with assms show ?thesis
+  by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that 
+        wq_es_cs' wq_s_cs) 
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) th' cs'"
+  obtains "th' \<noteq> th" "waiting s th' cs'"
+     |  "th' = th" "cs' = cs"
+proof(cases "waiting s th' cs'")
+  case True
+  have "th' \<noteq> th"
+  proof
+    assume otherwise: "th' = th"
+    from True[unfolded this]
+    show False by (simp add: th_not_waiting) 
+  qed
+  from that(1)[OF this True] show ?thesis .
+next
+  case False
+  hence "th' = th \<and> cs' = cs"
+      by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) 
+        set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
+  with that(2) show ?thesis by metis
+qed
+
+lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R" 
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:waiting_esE)
+      case 1
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case 2
+      thus ?thesis using waiting(1,2) by auto
+    qed
+  next
+    case (holding th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:holding_esE)
+      case 1
+      with holding(1,2)
+      show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  qed
+next
+  fix n1 n2
+  assume "(n1, n2) \<in> ?R"
+  hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
+  thus "(n1, n2) \<in> ?L"
+  proof
+    assume "(n1, n2) \<in> RAG s"
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from waiting_kept[OF this(3)]
+      show ?thesis using waiting(1,2)
+         by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from holding_kept[OF this(3)]
+      show ?thesis using holding(1,2)
+         by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  next
+    assume "n1 = Th th \<and> n2 = Cs cs"
+    thus ?thesis using RAG_edge by auto
+  qed
+qed
+
+end
+
+context valid_trace_p_h
+begin
+
+lemma wq_es_cs': "wq (e#s) cs = [th]"
+  using wq_es_cs[unfolded we] by simp
+
+lemma holding_es_th_cs: 
+  shows "holding (e#s) th cs"
+proof -
+  from wq_es_cs'
+  have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto
+  thus ?thesis using cs_holding_def holding_eq by blast 
+qed
+
+lemma RAG_edge: "(Cs cs, Th th) \<in> RAG (e#s)"
+  by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto)
+
+lemma waiting_esE:
+  assumes "waiting (e#s) th' cs'"
+  obtains "waiting s th' cs'"
+  using assms
+  by (metis cs_waiting_def event.distinct(15) is_p list.sel(1) 
+        set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv)
+  
+lemma holding_esE:
+  assumes "holding (e#s) th' cs'"
+  obtains "cs' \<noteq> cs" "holding s th' cs'"
+    | "cs' = cs" "th' = th"
+proof(cases "cs' = cs")
+  case True
+  from held_unique[OF holding_es_th_cs assms[unfolded True]]
+  have "th' = th" by simp
+  from that(2)[OF True this] show ?thesis .
+next
+  case False
+  have "holding s th' cs'" using assms
+    using False cs_holding_def holding_eq by auto
+  from that(1)[OF False this] show ?thesis .
+qed
+
+lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Cs cs, Th th)}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R" 
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:waiting_esE)
+      case 1
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    qed
+  next
+    case (holding th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:holding_esE)
+      case 1
+      with holding(1,2)
+      show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+    next
+      case 2
+      with holding(1,2) show ?thesis by auto
+    qed
+  qed
+next
+  fix n1 n2
+  assume "(n1, n2) \<in> ?R"
+  hence "(n1, n2) \<in> RAG s \<or> (n1 = Cs cs \<and> n2 = Th th)" by auto
+  thus "(n1, n2) \<in> ?L"
+  proof
+    assume "(n1, n2) \<in> RAG s"
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from waiting_kept[OF this(3)]
+      show ?thesis using waiting(1,2)
+         by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from holding_kept[OF this(3)]
+      show ?thesis using holding(1,2)
+         by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  next
+    assume "n1 = Cs cs \<and> n2 = Th th"
+    with holding_es_th_cs
+    show ?thesis 
+      by (unfold s_RAG_def, fold holding_eq, auto)
+  qed
+qed
+
+end
+
+context valid_trace_p
+begin
+
+lemma RAG_es': "RAG (e # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+                                                  else RAG s \<union> {(Th th, Cs cs)})"
+proof(cases "wq s cs = []")
+  case True
+  interpret vt_p: valid_trace_p_h using True
+    by (unfold_locales, simp)
+  show ?thesis by (simp add: vt_p.RAG_es vt_p.we) 
+next
+  case False
+  interpret vt_p: valid_trace_p_w using False
+    by (unfold_locales, simp)
+  show ?thesis by (simp add: vt_p.RAG_es vt_p.wne) 
+qed
+
+end
+
+
+end