
diff -r 17305a85493d -r c495eb16beb6 CpsG_1.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/CpsG_1.thy	Wed Jan 27 19:28:42 2016 +0800
@@ -0,0 +1,4403 @@
+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
+
+(* I am going to use this file as a start point to retrofiting 
+   PIPBasics.thy, which is originally called CpsG.ghy *)
+
+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
+
+end
+
+
+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_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
+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 (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) 
+  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
+
+(* An aux lemma used later *) 
+lemma unique_minus:
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r"
+  and xz: "(x, z) \<in> r^+"
+  and neq: "y \<noteq> z"
+  shows "(y, z) \<in> r^+"
+proof -
+ from xz and neq show ?thesis
+ proof(induct)
+   case (base ya)
+   have "(x, ya) \<in> r" by fact
+   from unique [OF xy this] have "y = ya" .
+   with base show ?case by auto
+ next
+   case (step ya z)
+   show ?case
+   proof(cases "y = ya")
+     case True
+     from step True show ?thesis by simp
+   next
+     case False
+     from step False
+     show ?thesis by auto
+   qed
+ qed
+qed
+
+lemma unique_base:
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r"
+  and xz: "(x, z) \<in> r^+"
+  and neq_yz: "y \<noteq> z"
+  shows "(y, z) \<in> r^+"
+proof -
+  from xz neq_yz show ?thesis
+  proof(induct)
+    case (base ya)
+    from xy unique base show ?case by auto
+  next
+    case (step ya z)
+    show ?case
+    proof(cases "y = ya")
+      case True
+      from True step show ?thesis by auto
+    next
+      case False
+      from False step 
+      have "(y, ya) \<in> r\<^sup>+" by auto
+      with step show ?thesis by auto
+    qed
+  qed
+qed
+
+lemma unique_chain:
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r^+"
+  and xz: "(x, z) \<in> r^+"
+  and neq_yz: "y \<noteq> z"
+  shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
+proof -
+  from xy xz neq_yz show ?thesis
+  proof(induct)
+    case (base y)
+    have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
+    from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
+  next
+    case (step y za)
+    show ?case
+    proof(cases "y = z")
+      case True
+      from True step show ?thesis by auto
+    next
+      case False
+      from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
+      thus ?thesis
+      proof
+        assume "(z, y) \<in> r\<^sup>+"
+        with step have "(z, za) \<in> r\<^sup>+" by auto
+        thus ?thesis by auto
+      next
+        assume h: "(y, z) \<in> r\<^sup>+"
+        from step have yza: "(y, za) \<in> r" by simp
+        from step have "za \<noteq> z" by simp
+        from unique_minus [OF _ yza h this] and unique
+        have "(za, z) \<in> r\<^sup>+" by auto
+        thus ?thesis by auto
+      qed
+    qed
+  qed
+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)
+
+definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis by (cases, simp)
+qed
+
+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 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 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_n
+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 (* ccc *)
+
+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
+
+
+
+context valid_trace
+begin
+
+lemma finite_threads:
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+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)
+
+lemma wq_threads: 
+  assumes h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+
+
+lemma wq_threads: 
+  assumes h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+proof -
+ from vt and h show ?thesis
+  proof(induct arbitrary: th cs)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s
+      using vt_cons(1) by (unfold_locales, auto)
+    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+      and stp: "step s e"
+      and vt: "vt s"
+      and h: "th \<in> set (wq (e # s) cs)"
+    show ?case
+    proof(cases e)
+      case (Create th' prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    next
+      case (Exit th')
+      with stp ih h show ?thesis
+        apply (auto simp:wq_def Let_def)
+        apply (ind_cases "step s (Exit th')")
+        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
+               s_RAG_def s_holding_def cs_holding_def)
+        done
+    next
+      case (V th' cs')
+      show ?thesis
+      proof(cases "cs' = cs")
+        case False
+        with h
+        show ?thesis
+          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+          by (drule_tac ih, simp)
+      next
+        case True
+        from h
+        show ?thesis
+        proof(unfold V wq_def)
+          assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+          show "th \<in> threads (V th' cs' # s)"
+          proof(cases "cs = cs'")
+            case False
+            hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
+            with th_in have " th \<in> set (wq s cs)" 
+              by (fold wq_def, simp)
+            from ih [OF this] show ?thesis by simp
+          next
+            case True
+            show ?thesis
+            proof(cases "wq_fun (schs s) cs'")
+              case Nil
+              with h V show ?thesis
+                apply (auto simp:wq_def Let_def split:if_splits)
+                by (fold wq_def, drule_tac ih, simp)
+            next
+              case (Cons a rest)
+              assume eq_wq: "wq_fun (schs s) cs' = a # rest"
+              with h V show ?thesis
+                apply (auto simp:Let_def wq_def split:if_splits)
+              proof -
+                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
+                proof(rule someI2)
+                  from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
+                  show "distinct rest \<and> set rest = set rest" by auto
+                next
+                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+                    by auto
+                qed
+                with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
+                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+              next
+                assume th_in: "th \<in> set (wq_fun (schs s) cs)"
+                from ih[OF this[folded wq_def]]
+                show "th \<in> threads s" .
+              qed
+            qed
+          qed
+        qed
+      qed
+    next
+      case (P th' cs')
+      from h stp
+      show ?thesis
+        apply (unfold P wq_def)
+        apply (auto simp:Let_def split:if_splits, fold wq_def)
+        apply (auto intro:ih)
+        apply(ind_cases "step s (P th' cs')")
+        by (unfold runing_def readys_def, auto)
+    next
+      case (Set thread prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    qed
+  next
+    case vt_nil
+    thus ?case by (auto simp:wq_def)
+  qed
+qed
+
+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 le_cp:
+  shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+  show "Prc (priority th s) (last_set th s)
+    \<le> Max (insert (Prc (priority th s) (last_set th s))
+            ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
+    (is "?l \<le> Max (insert ?l ?A)")
+  proof(cases "?A = {}")
+    case False
+    have "finite ?A" (is "finite (?f ` ?B)")
+    proof -
+      have "finite ?B" 
+      proof-
+        have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
+        proof -
+          let ?F = "\<lambda> (x, y). the_th x"
+          have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+            apply (auto simp:image_def)
+            by (rule_tac x = "(Th x, Th th)" in bexI, auto)
+          moreover have "finite \<dots>"
+          proof -
+            from finite_RAG have "finite (RAG s)" .
+            hence "finite ((RAG (wq s))\<^sup>+)"
+              apply (unfold finite_trancl)
+              by (auto simp: s_RAG_def cs_RAG_def wq_def)
+            thus ?thesis by auto
+          qed
+          ultimately show ?thesis by (auto intro:finite_subset)
+        qed
+        thus ?thesis by (simp add:cs_dependants_def)
+      qed
+      thus ?thesis by simp
+    qed
+    from Max_insert [OF this False, of ?l] show ?thesis by auto
+  next
+    case True
+    thus ?thesis by auto
+  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'" by auto
+    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 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 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
+
+
+
+
+
+text {* (* ddd *) 
+  The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
+  with the happening of @{text "V"}-events:
+*}
+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'}"
+  apply (insert vt, unfold s_RAG_def) 
+  apply (auto split:if_splits list.splits simp:Let_def)
+  apply (auto elim: step_v_waiting_mono step_v_hold_inv
+              step_v_release step_v_wait_inv
+              step_v_get_hold step_v_release_inv)
+  apply (erule_tac step_v_not_wait, auto)
+  done
+
+text {* 
+  The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
+  with the happening of @{text "P"}-events:
+*}
+lemma step_RAG_p:
+  "vt (P th cs#s) \<Longrightarrow>
+  RAG (P th cs # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+                                             else RAG s \<union> {(Th th, Cs cs)})"
+  apply(simp only: s_RAG_def wq_def)
+  apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
+  apply(case_tac "csa = cs", auto)
+  apply(fold wq_def)
+  apply(drule_tac step_back_step)
+  apply(ind_cases " step s (P (hd (wq s cs)) cs)")
+  apply(simp add:s_RAG_def wq_def cs_holding_def)
+  apply(auto)
+  done
+
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, auto)
+
+context valid_trace
+begin
+
+text {*
+  The following lemma shows that @{text "RAG"} is acyclic.
+  The overall structure is by induction on the formation of @{text "vt s"}
+  and then case analysis on event @{text "e"}, where the non-trivial cases 
+  for those for @{text "V"} and @{text "P"} events.
+*}
+lemma acyclic_RAG:
+  shows "acyclic (RAG s)"
+using vt
+proof(induct)
+  case (vt_cons s e)
+  interpret vt_s: valid_trace s using vt_cons(1)
+    by (unfold_locales, simp)
+  assume ih: "acyclic (RAG s)"
+    and stp: "step s e"
+    and vt: "vt s"
+  show ?case
+  proof(cases e)
+    case (Create th prio)
+    with ih
+    show ?thesis by (simp add:RAG_create_unchanged)
+  next
+    case (Exit th)
+    with ih show ?thesis by (simp add:RAG_exit_unchanged)
+  next
+    case (V th cs)
+    from V vt stp have vtt: "vt (V th cs#s)" by auto
+    from step_RAG_v [OF this]
+    have eq_de: 
+      "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 = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+    from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+    from step_back_step [OF vtt]
+    have "step s (V th cs)" .
+    thus ?thesis
+    proof(cases)
+      assume "holding s th cs"
+      hence th_in: "th \<in> set (wq s cs)" and
+        eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
+      then obtain rest where
+        eq_wq: "wq s cs = th#rest"
+        by (cases "wq s cs", auto)
+      show ?thesis
+      proof(cases "rest = []")
+        case False
+        let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+        from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
+          by (unfold next_th_def, auto)
+        let ?E = "(?A - ?B - ?C)"
+        have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
+        proof
+          assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
+          hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+          from tranclD [OF this]
+          obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
+          hence th_d: "(Th ?th', x) \<in> ?A" by simp
+          from RAG_target_th [OF this]
+          obtain cs' where eq_x: "x = Cs cs'" by auto
+          with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
+          hence wt_th': "waiting s ?th' cs'"
+            unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
+          hence "cs' = cs"
+          proof(rule vt_s.waiting_unique)
+            from eq_wq vt_s.wq_distinct[of cs]
+            show "waiting s ?th' cs" 
+              apply (unfold s_waiting_def wq_def, auto)
+            proof -
+              assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq_fun (schs s) cs = th # rest"
+              have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+              next
+                fix x assume "distinct x \<and> set x = set rest"
+                with False show "x \<noteq> []" by auto
+              qed
+              hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
+                set (SOME q. distinct q \<and> set q = set rest)" by auto
+              moreover have "\<dots> = set rest" 
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+              next
+                show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+              qed
+              moreover note hd_in
+              ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
+            next
+              assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+              have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" by auto
+              next
+                fix x assume "distinct x \<and> set x = set rest"
+                with False show "x \<noteq> []" by auto
+              qed
+              hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
+                set (SOME q. distinct q \<and> set q = set rest)" by auto
+              moreover have "\<dots> = set rest" 
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" by auto
+              next
+                show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+              qed
+              moreover note hd_in
+              ultimately show False by auto
+            qed
+          qed
+          with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
+          with False
+          show "False" by (auto simp: next_th_def eq_wq)
+        qed
+        with acyclic_insert[symmetric] and ac
+          and eq_de eq_D show ?thesis by auto
+      next
+        case True
+        with eq_wq
+        have eq_D: "?D = {}"
+          by (unfold next_th_def, auto)
+        with eq_de ac
+        show ?thesis by auto
+      qed 
+    qed
+  next
+    case (P th cs)
+    from P vt stp have vtt: "vt (P th cs#s)" by auto
+    from step_RAG_p [OF this] P
+    have "RAG (e # s) = 
+      (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else 
+      RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+      by simp
+    moreover have "acyclic ?R"
+    proof(cases "wq s cs = []")
+      case True
+      hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
+      have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
+      proof
+        assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
+        hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+        from tranclD2 [OF this]
+        obtain x where "(x, Cs cs) \<in> RAG s" by auto
+        with True show False by (auto simp:s_RAG_def cs_waiting_def)
+      qed
+      with acyclic_insert ih eq_r show ?thesis by auto
+    next
+      case False
+      hence eq_r: "?R =  RAG s \<union> {(Th th, Cs cs)}" by simp
+      have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
+      proof
+        assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
+        hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+        moreover from step_back_step [OF vtt] have "step s (P th cs)" .
+        ultimately show False
+        proof -
+          show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+            by (ind_cases "step s (P th cs)", simp)
+        qed
+      qed
+      with acyclic_insert ih eq_r show ?thesis by auto
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (Set thread prio)
+      with ih
+      thm RAG_set_unchanged
+      show ?thesis by (simp add:RAG_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "acyclic (RAG ([]::state))"
+      by (auto simp: s_RAG_def cs_waiting_def 
+        cs_holding_def wq_def acyclic_def)
+qed
+
+
+lemma finite_RAG:
+  shows "finite (RAG s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s using vt_cons(1)
+      by (unfold_locales, simp)
+    assume ih: "finite (RAG s)"
+      and stp: "step s e"
+      and vt: "vt s"
+    show ?case
+    proof(cases e)
+      case (Create th prio)
+      with ih
+      show ?thesis by (simp add:RAG_create_unchanged)
+    next
+      case (Exit th)
+      with ih show ?thesis by (simp add:RAG_exit_unchanged)
+    next
+      case (V th cs)
+      from V vt stp have vtt: "vt (V th cs#s)" by auto
+      from step_RAG_v [OF this]
+      have eq_de: "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 = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+      moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
+      moreover have "finite ?D"
+      proof -
+        have "?D = {} \<or> (\<exists> a. ?D = {a})" 
+          by (unfold next_th_def, auto)
+        thus ?thesis
+        proof
+          assume h: "?D = {}"
+          show ?thesis by (unfold h, simp)
+        next
+          assume "\<exists> a. ?D = {a}"
+          thus ?thesis
+            by (metis finite.simps)
+        qed
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (P th cs)
+      from P vt stp have vtt: "vt (P th cs#s)" by auto
+      from step_RAG_p [OF this] P
+      have "RAG (e # s) = 
+              (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else 
+                                    RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+        by simp
+      moreover have "finite ?R"
+      proof(cases "wq s cs = []")
+        case True
+        hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
+        with True and ih show ?thesis by auto
+      next
+        case False
+        hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
+        with False and ih show ?thesis by auto
+      qed
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio)
+      with ih
+      show ?thesis by (simp add:RAG_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "finite (RAG ([]::state))"
+      by (auto simp: s_RAG_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+text {* Several useful lemmas *}
+
+lemma wf_dep_converse: 
+  shows "wf ((RAG s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+  from finite_RAG 
+  show "finite (RAG s)" .
+next
+  from acyclic_RAG
+  show "acyclic (RAG s)" .
+qed
+
+end
+
+lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
+  by (induct l, auto)
+
+lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
+  by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+
+context valid_trace
+begin
+
+lemma wq_threads: 
+  assumes h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+proof -
+ from vt and h show ?thesis
+  proof(induct arbitrary: th cs)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s
+      using vt_cons(1) by (unfold_locales, auto)
+    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+      and stp: "step s e"
+      and vt: "vt s"
+      and h: "th \<in> set (wq (e # s) cs)"
+    show ?case
+    proof(cases e)
+      case (Create th' prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    next
+      case (Exit th')
+      with stp ih h show ?thesis
+        apply (auto simp:wq_def Let_def)
+        apply (ind_cases "step s (Exit th')")
+        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
+               s_RAG_def s_holding_def cs_holding_def)
+        done
+    next
+      case (V th' cs')
+      show ?thesis
+      proof(cases "cs' = cs")
+        case False
+        with h
+        show ?thesis
+          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+          by (drule_tac ih, simp)
+      next
+        case True
+        from h
+        show ?thesis
+        proof(unfold V wq_def)
+          assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+          show "th \<in> threads (V th' cs' # s)"
+          proof(cases "cs = cs'")
+            case False
+            hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
+            with th_in have " th \<in> set (wq s cs)" 
+              by (fold wq_def, simp)
+            from ih [OF this] show ?thesis by simp
+          next
+            case True
+            show ?thesis
+            proof(cases "wq_fun (schs s) cs'")
+              case Nil
+              with h V show ?thesis
+                apply (auto simp:wq_def Let_def split:if_splits)
+                by (fold wq_def, drule_tac ih, simp)
+            next
+              case (Cons a rest)
+              assume eq_wq: "wq_fun (schs s) cs' = a # rest"
+              with h V show ?thesis
+                apply (auto simp:Let_def wq_def split:if_splits)
+              proof -
+                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
+                proof(rule someI2)
+                  from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
+                  show "distinct rest \<and> set rest = set rest" by auto
+                next
+                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+                    by auto
+                qed
+                with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
+                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+              next
+                assume th_in: "th \<in> set (wq_fun (schs s) cs)"
+                from ih[OF this[folded wq_def]]
+                show "th \<in> threads s" .
+              qed
+            qed
+          qed
+        qed
+      qed
+    next
+      case (P th' cs')
+      from h stp
+      show ?thesis
+        apply (unfold P wq_def)
+        apply (auto simp:Let_def split:if_splits, fold wq_def)
+        apply (auto intro:ih)
+        apply(ind_cases "step s (P th' cs')")
+        by (unfold runing_def readys_def, auto)
+    next
+      case (Set thread prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    qed
+  next
+    case vt_nil
+    thus ?case by (auto simp:wq_def)
+  qed
+qed
+
+lemma range_in: "\<lbrakk>(Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+  apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
+  by (auto intro:wq_threads)
+
+lemma readys_v_eq:
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and not_in: "th \<notin>  set rest"
+  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
+proof -
+  from assms show ?thesis
+    apply (auto simp:readys_def)
+    apply(simp add:s_waiting_def[folded wq_def])
+    apply (erule_tac x = csa in allE)
+    apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
+    apply (case_tac "csa = cs", simp)
+    apply (erule_tac x = cs in allE)
+    apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
+    apply(auto simp add: wq_def)
+    apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
+    proof -
+       assume th_nin: "th \<notin> set rest"
+        and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+        and eq_wq: "wq_fun (schs s) cs = thread # rest"
+      have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+      proof(rule someI2)
+        from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
+        show "distinct rest \<and> set rest = set rest" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+      qed
+      with th_nin th_in show False by auto
+    qed
+qed
+
+text {* \noindent
+  The following lemmas shows that: starting from any node in @{text "RAG"}, 
+  by chasing out-going edges, it is always possible to reach a node representing a ready
+  thread. In this lemma, it is the @{text "th'"}.
+*}
+
+lemma chain_building:
+  shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
+proof -
+  from wf_dep_converse
+  have h: "wf ((RAG s)\<inverse>)" .
+  show ?thesis
+  proof(induct rule:wf_induct [OF h])
+    fix x
+    assume ih [rule_format]: 
+      "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow> 
+           y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
+    show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
+    proof
+      assume x_d: "x \<in> Domain (RAG s)"
+      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
+      proof(cases x)
+        case (Th th)
+        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
+        with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
+        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
+        hence "Cs cs \<in> Domain (RAG s)" by auto
+        from ih [OF x_in_r this] obtain th'
+          where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
+        have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
+        with th'_ready show ?thesis by auto
+      next
+        case (Cs cs)
+        from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
+        show ?thesis
+        proof(cases "th' \<in> readys s")
+          case True
+          from True and th'_d show ?thesis by auto
+        next
+          case False
+          from th'_d and range_in  have "th' \<in> threads s" by auto
+          with False have "Th th' \<in> Domain (RAG s)" 
+            by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
+          from ih [OF th'_d this]
+          obtain th'' where 
+            th''_r: "th'' \<in> readys s" and 
+            th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
+          from th'_d and th''_in 
+          have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
+          with th''_r show ?thesis by auto
+        qed
+      qed
+    qed
+  qed
+qed
+
+text {* \noindent
+  The following is just an instance of @{text "chain_building"}.
+*}
+lemma th_chain_to_ready:
+  assumes th_in: "th \<in> threads s"
+  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
+proof(cases "th \<in> readys s")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  from False and th_in have "Th th \<in> Domain (RAG s)" 
+    by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
+  from chain_building [rule_format, OF this]
+  show ?thesis by auto
+qed
+
+end
+
+
+
+lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
+  by (unfold s_holding_def cs_holding_def, auto)
+
+context valid_trace
+begin
+
+lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
+  apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
+  by(auto elim:waiting_unique holding_unique)
+
+end
+
+
+lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
+by (induct rule:trancl_induct, auto)
+
+context valid_trace
+begin
+
+lemma dchain_unique:
+  assumes th1_d: "(n, Th th1) \<in> (RAG s)^+"
+  and th1_r: "th1 \<in> readys s"
+  and th2_d: "(n, Th th2) \<in> (RAG s)^+"
+  and th2_r: "th2 \<in> readys s"
+  shows "th1 = th2"
+proof -
+  { assume neq: "th1 \<noteq> th2"
+    hence "Th th1 \<noteq> Th th2" by simp
+    from unique_chain [OF _ th1_d th2_d this] and unique_RAG 
+    have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
+    hence "False"
+    proof
+      assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+      from dd eq_n have "th1 \<notin> readys s"
+        by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
+      with th1_r show ?thesis by auto
+    next
+      assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+      from dd eq_n have "th2 \<notin> readys s"
+        by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
+      with th2_r show ?thesis by auto
+    qed
+  } thus ?thesis by auto
+qed
+
+end
+             
+
+lemma step_holdents_p_add:
+  assumes vt: "vt (P th cs#s)"
+  and "wq s cs = []"
+  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from assms show ?thesis
+  unfolding  holdents_test step_RAG_p[OF vt] by (auto)
+qed
+
+lemma step_holdents_p_eq:
+  assumes vt: "vt (P th cs#s)"
+  and "wq s cs \<noteq> []"
+  shows "holdents (P th cs#s) th = holdents s th"
+proof -
+  from assms show ?thesis
+  unfolding  holdents_test step_RAG_p[OF vt] by auto
+qed
+
+
+lemma (in valid_trace) finite_holding :
+  shows "finite (holdents s th)"
+proof -
+  let ?F = "\<lambda> (x, y). the_cs x"
+  from finite_RAG 
+  have "finite (RAG s)" .
+  hence "finite (?F `(RAG s))" by simp
+  moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>" 
+  proof -
+    { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
+      fix x assume "(Cs x, Th th) \<in> RAG s"
+      hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
+      moreover have "?F (Cs x, Th th) = x" by simp
+      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp 
+    } thus ?thesis by auto
+  qed
+  ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
+qed
+
+lemma cntCS_v_dec: 
+  assumes vtv: "vt (V thread cs#s)"
+  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
+proof -
+  from vtv interpret vt_s: valid_trace s
+    by (cases, unfold_locales, simp)
+  from vtv interpret vt_v: valid_trace "V thread cs#s"
+     by (unfold_locales, simp)
+  from step_back_step[OF vtv]
+  have cs_in: "cs \<in> holdents s thread" 
+    apply (cases, unfold holdents_test s_RAG_def, simp)
+    by (unfold cs_holding_def s_holding_def wq_def, auto)
+  moreover have cs_not_in: 
+    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
+    apply (insert vt_s.wq_distinct[of cs])
+    apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
+            auto simp:next_th_def)
+  proof -
+    fix rest
+    assume dst: "distinct (rest::thread list)"
+      and ne: "rest \<noteq> []"
+    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+    qed
+    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                     set (SOME q. distinct q \<and> set q = set rest)" by simp
+    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      fix x assume " distinct x \<and> set x = set rest" with ne
+      show "x \<noteq> []" by auto
+    qed
+    ultimately 
+    show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
+      by auto
+  next
+    fix rest
+    assume dst: "distinct (rest::thread list)"
+      and ne: "rest \<noteq> []"
+    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+    qed
+    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                     set (SOME q. distinct q \<and> set q = set rest)" by simp
+    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      fix x assume " distinct x \<and> set x = set rest" with ne
+      show "x \<noteq> []" by auto
+    qed
+    ultimately show "False" by auto 
+  qed
+  ultimately 
+  have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
+    by auto
+  moreover have "card \<dots> = 
+                    Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
+  proof(rule card_insert)
+    from vt_v.finite_holding
+    show " finite (holdents (V thread cs # s) thread)" .
+  qed
+  moreover from cs_not_in 
+  have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
+  ultimately show ?thesis by (simp add:cntCS_def)
+qed 
+
+lemma count_rec1 [simp]: 
+  assumes "Q e"
+  shows "count Q (e#es) = Suc (count Q es)"
+  using assms
+  by (unfold count_def, auto)
+
+lemma count_rec2 [simp]: 
+  assumes "\<not>Q e"
+  shows "count Q (e#es) = (count Q es)"
+  using assms
+  by (unfold count_def, auto)
+
+lemma count_rec3 [simp]: 
+  shows "count Q [] =  0"
+  by (unfold count_def, auto)
+
+lemma cntP_diff_inv:
+  assumes "cntP (e#s) th \<noteq> cntP s th"
+  shows "isP e \<and> actor e = th"
+proof(cases e)
+  case (P th' pty)
+  show ?thesis
+  by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)", 
+        insert assms P, auto simp:cntP_def)
+qed (insert assms, auto simp:cntP_def)
+  
+lemma cntV_diff_inv:
+  assumes "cntV (e#s) th \<noteq> cntV s th"
+  shows "isV e \<and> actor e = th"
+proof(cases e)
+  case (V th' pty)
+  show ?thesis
+  by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)", 
+        insert assms V, auto simp:cntV_def)
+qed (insert assms, auto simp:cntV_def)
+
+context valid_trace
+begin
+
+text {* (* ddd *) \noindent
+  The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} 
+  of one particular thread. t
+*} 
+
+lemma cnp_cnv_cncs:
+  shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) 
+                                       then cntCS s th else cntCS s th + 1)"
+proof -
+  from vt show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
+    assume vt: "vt s"
+    and ih: "\<And>th. cntP s th  = cntV s th +
+               (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
+    and stp: "step s e"
+    from stp show ?case
+    proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+        and not_in: "thread \<notin> threads s"
+      show ?thesis
+      proof -
+        { fix cs 
+          assume "thread \<in> set (wq s cs)"
+          from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
+          with not_in have "False" by simp
+        } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
+          by (auto simp:readys_def threads.simps s_waiting_def 
+            wq_def cs_waiting_def Let_def)
+        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+        have eq_cncs: "cntCS (e#s) th = cntCS s th"
+          unfolding cntCS_def holdents_test
+          by (simp add:RAG_create_unchanged eq_e)
+        { assume "th \<noteq> thread"
+          with eq_readys eq_e
+          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+            by (simp add:threads.simps)
+          with eq_cnp eq_cnv eq_cncs ih not_in
+          have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp
+          moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
+          moreover note eq_cnp eq_cnv eq_cncs
+          ultimately have ?thesis by auto
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread" 
+      and is_runing: "thread \<in> runing s"
+      and no_hold: "holdents s thread = {}"
+      from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+      from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+      have eq_cncs: "cntCS (e#s) th = cntCS s th"
+        unfolding cntCS_def holdents_test
+        by (simp add:RAG_exit_unchanged eq_e)
+      { assume "th \<noteq> thread"
+        with eq_e
+        have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+          (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+          apply (simp add:threads.simps readys_def)
+          apply (subst s_waiting_def)
+          apply (simp add:Let_def)
+          apply (subst s_waiting_def, simp)
+          done
+        with eq_cnp eq_cnv eq_cncs ih
+        have ?thesis by simp
+      } moreover {
+        assume eq_th: "th = thread"
+        with ih is_runing have " cntP s th = cntV s th + cntCS s th" 
+          by (simp add:runing_def)
+        moreover from eq_th eq_e have "th \<notin> threads (e#s)"
+          by simp
+        moreover note eq_cnp eq_cnv eq_cncs
+        ultimately have ?thesis by auto
+      } ultimately show ?thesis by blast
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+        and is_runing: "thread \<in> runing s"
+        and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
+      from thread_P vt stp ih  have vtp: "vt (P thread cs#s)" by auto
+      then interpret vt_p: valid_trace "(P thread cs#s)"
+        by (unfold_locales, simp)
+      show ?thesis 
+      proof -
+        { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
+          assume neq_th: "th \<noteq> thread"
+          with eq_e
+          have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
+            apply (simp add:readys_def s_waiting_def wq_def Let_def)
+            apply (rule_tac hh)
+             apply (intro iffI allI, clarify)
+            apply (erule_tac x = csa in allE, auto)
+            apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
+            apply (erule_tac x = cs in allE, auto)
+            by (case_tac "(wq_fun (schs s) cs)", auto)
+          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+            apply (simp add:cntCS_def holdents_test)
+            by (unfold  step_RAG_p [OF vtp], auto)
+          moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
+            by (simp add:cntP_def count_def)
+          moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
+            by (simp add:cntV_def count_def)
+          moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
+          moreover note ih [of th] 
+          ultimately have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          have ?thesis
+          proof -
+            from eq_e eq_th have eq_cnp: "cntP (e # s) th  = 1 + (cntP s th)" 
+              by (simp add:cntP_def count_def)
+            from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
+              by (simp add:cntV_def count_def)
+            show ?thesis
+            proof (cases "wq s cs = []")
+              case True
+              with is_runing
+              have "th \<in> readys (e#s)"
+                apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
+                apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
+                by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
+              moreover have "cntCS (e # s) th = 1 + cntCS s th"
+              proof -
+                have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
+                  Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
+                proof -
+                  have "?L = insert cs ?R" by auto
+                  moreover have "card \<dots> = Suc (card (?R - {cs}))" 
+                  proof(rule card_insert)
+                    from vt_s.finite_holding [of thread]
+                    show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
+                      by (unfold holdents_test, simp)
+                  qed
+                  moreover have "?R - {cs} = ?R"
+                  proof -
+                    have "cs \<notin> ?R"
+                    proof
+                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
+                      with no_dep show False by auto
+                    qed
+                    thus ?thesis by auto
+                  qed
+                  ultimately show ?thesis by auto
+                qed
+                thus ?thesis
+                  apply (unfold eq_e eq_th cntCS_def)
+                  apply (simp add: holdents_test)
+                  by (unfold step_RAG_p [OF vtp], auto simp:True)
+              qed
+              moreover from is_runing have "th \<in> readys s"
+                by (simp add:runing_def eq_th)
+              moreover note eq_cnp eq_cnv ih [of th]
+              ultimately show ?thesis by auto
+            next
+              case False
+              have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
+                    by (unfold eq_th eq_e wq_def, auto simp:Let_def)
+              have "th \<notin> readys (e#s)"
+              proof
+                assume "th \<in> readys (e#s)"
+                hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
+                from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
+                hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)" 
+                  by (simp add:s_waiting_def wq_def)
+                moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
+                ultimately have "th = hd (wq (e#s) cs)" by blast
+                with eq_wq have "th = hd (wq s cs @ [th])" by simp
+                hence "th = hd (wq s cs)" using False by auto
+                with False eq_wq vt_p.wq_distinct [of cs]
+                show False by (fold eq_e, auto)
+              qed
+              moreover from is_runing have "th \<in> threads (e#s)" 
+                by (unfold eq_e, auto simp:runing_def readys_def eq_th)
+              moreover have "cntCS (e # s) th = cntCS s th"
+                apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
+                by (auto simp:False)
+              moreover note eq_cnp eq_cnv ih[of th]
+              moreover from is_runing have "th \<in> readys s"
+                by (simp add:runing_def eq_th)
+              ultimately show ?thesis by auto
+            qed
+          qed
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_V thread cs)
+      from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
+      then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
+      assume eq_e: "e = V thread cs"
+        and is_runing: "thread \<in> runing s"
+        and hold: "holding s thread cs"
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+      have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+      proof(rule someI2)
+        from vt_v.wq_distinct[of cs] and eq_wq
+        show "distinct rest \<and> set rest = set rest"
+          by (metis distinct.simps(2) vt_s.wq_distinct)
+      next
+        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+          by auto
+      qed
+      show ?thesis
+      proof -
+        { assume eq_th: "th = thread"
+          from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
+            by (unfold eq_e, simp add:cntP_def count_def)
+          moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
+            by (unfold eq_e, simp add:cntV_def count_def)
+          moreover from cntCS_v_dec [OF vtv] 
+          have "cntCS (e # s) thread + 1 = cntCS s thread"
+            by (simp add:eq_e)
+          moreover from is_runing have rd_before: "thread \<in> readys s"
+            by (unfold runing_def, simp)
+          moreover have "thread \<in> readys (e # s)"
+          proof -
+            from is_runing
+            have "thread \<in> threads (e#s)" 
+              by (unfold eq_e, auto simp:runing_def readys_def)
+            moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
+            proof
+              fix cs1
+              { assume eq_cs: "cs1 = cs" 
+                have "\<not> waiting (e # s) thread cs1"
+                proof -
+                  from eq_wq
+                  have "thread \<notin> set (wq (e#s) cs1)"
+                    apply(unfold eq_e wq_def eq_cs s_holding_def)
+                    apply (auto simp:Let_def)
+                  proof -
+                    assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                    with eq_set have "thread \<in> set rest" by simp
+                    with vt_v.wq_distinct[of cs]
+                    and eq_wq show False
+                        by (metis distinct.simps(2) vt_s.wq_distinct)
+                  qed
+                  thus ?thesis by (simp add:wq_def s_waiting_def)
+                qed
+              } moreover {
+                assume neq_cs: "cs1 \<noteq> cs"
+                  have "\<not> waiting (e # s) thread cs1" 
+                  proof -
+                    from wq_v_neq [OF neq_cs[symmetric]]
+                    have "wq (V thread cs # s) cs1 = wq s cs1" .
+                    moreover have "\<not> waiting s thread cs1" 
+                    proof -
+                      from runing_ready and is_runing
+                      have "thread \<in> readys s" by auto
+                      thus ?thesis by (simp add:readys_def)
+                    qed
+                    ultimately show ?thesis 
+                      by (auto simp:wq_def s_waiting_def eq_e)
+                  qed
+              } ultimately show "\<not> waiting (e # s) thread cs1" by blast
+            qed
+            ultimately show ?thesis by (simp add:readys_def)
+          qed
+          moreover note eq_th ih
+          ultimately have ?thesis by auto
+        } moreover {
+          assume neq_th: "th \<noteq> thread"
+          from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" 
+            by (simp add:cntP_def count_def)
+          from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" 
+            by (simp add:cntV_def count_def)
+          have ?thesis
+          proof(cases "th \<in> set rest")
+            case False
+            have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+              apply (insert step_back_vt[OF vtv])
+              by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
+            moreover have "cntCS (e#s) th = cntCS s th"
+              apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
+              proof -
+                have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+                      {cs. (Cs cs, Th th) \<in> RAG s}"
+                proof -
+                  from False eq_wq
+                  have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
+                    apply (unfold next_th_def, auto)
+                  proof -
+                    assume ne: "rest \<noteq> []"
+                      and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                      and eq_wq: "wq s cs = thread # rest"
+                    from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                                  set (SOME q. distinct q \<and> set q = set rest)
+                                  " by simp
+                    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+                    proof(rule someI2)
+                      from vt_s.wq_distinct[ of cs] and eq_wq
+                      show "distinct rest \<and> set rest = set rest" by auto
+                    next
+                      fix x assume "distinct x \<and> set x = set rest"
+                      with ne show "x \<noteq> []" by auto
+                    qed
+                    ultimately show 
+                      "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
+                      by auto
+                  qed    
+                  thus ?thesis by auto
+                qed
+                thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+                             card {cs. (Cs cs, Th th) \<in> RAG s}" by simp 
+              qed
+            moreover note ih eq_cnp eq_cnv eq_threads
+            ultimately show ?thesis by auto
+          next
+            case True
+            assume th_in: "th \<in> set rest"
+            show ?thesis
+            proof(cases "next_th s thread cs th")
+              case False
+              with eq_wq and th_in have 
+                neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
+                by (auto simp:next_th_def)
+              have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+              proof -
+                from eq_wq and th_in
+                have "\<not> th \<in> readys s"
+                  apply (auto simp:readys_def s_waiting_def)
+                  apply (rule_tac x = cs in exI, auto)
+                  by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
+                moreover 
+                from eq_wq and th_in and neq_hd
+                have "\<not> (th \<in> readys (e # s))"
+                  apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
+                  by (rule_tac x = cs in exI, auto simp:eq_set)
+                ultimately show ?thesis by auto
+              qed
+              moreover have "cntCS (e#s) th = cntCS s th" 
+              proof -
+                from eq_wq and  th_in and neq_hd
+                have "(holdents (e # s) th) = (holdents s th)"
+                  apply (unfold eq_e step_RAG_v[OF vtv], 
+                         auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
+                                   Let_def cs_holding_def)
+                  by (insert vt_s.wq_distinct[of cs], auto simp:wq_def)
+                thus ?thesis by (simp add:cntCS_def)
+              qed
+              moreover note ih eq_cnp eq_cnv eq_threads
+              ultimately show ?thesis by auto
+            next
+              case True
+              let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
+              let ?t = "hd ?rest"
+              from True eq_wq th_in neq_th
+              have "th \<in> readys (e # s)"
+                apply (auto simp:eq_e readys_def s_waiting_def wq_def
+                        Let_def next_th_def)
+              proof -
+                assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                show "?t \<in> threads s"
+                proof(rule vt_s.wq_threads)
+                  from eq_wq and t_in
+                  show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
+                qed
+              next
+                fix csa
+                assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                  and neq_cs: "csa \<noteq> cs"
+                  and t_in': "?t \<in>  set (wq_fun (schs s) csa)"
+                show "?t = hd (wq_fun (schs s) csa)"
+                proof -
+                  { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
+                    from vt_s.wq_distinct[of cs] and 
+                    eq_wq[folded wq_def] and t_in eq_wq
+                    have "?t \<noteq> thread" by auto
+                    with eq_wq and t_in
+                    have w1: "waiting s ?t cs"
+                      by (auto simp:s_waiting_def wq_def)
+                    from t_in' neq_hd'
+                    have w2: "waiting s ?t csa"
+                      by (auto simp:s_waiting_def wq_def)
+                    from vt_s.waiting_unique[OF w1 w2]
+                    and neq_cs have "False" by auto
+                  } thus ?thesis by auto
+                qed
+              qed
+              moreover have "cntP s th = cntV s th + cntCS s th + 1"
+              proof -
+                have "th \<notin> readys s" 
+                proof -
+                  from True eq_wq neq_th th_in
+                  show ?thesis
+                    apply (unfold readys_def s_waiting_def, auto)
+                    by (rule_tac x = cs in exI, auto simp add: wq_def)
+                qed
+                moreover have "th \<in> threads s"
+                proof -
+                  from th_in eq_wq
+                  have "th \<in> set (wq s cs)" by simp
+                  from vt_s.wq_threads [OF this] 
+                  show ?thesis .
+                qed
+                ultimately show ?thesis using ih by auto
+              qed
+              moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
+                apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
+              proof -
+                show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
+                               Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
+                  (is "card ?A = Suc (card ?B)")
+                proof -
+                  have "?A = insert cs ?B" by auto
+                  hence "card ?A = card (insert cs ?B)" by simp
+                  also have "\<dots> = Suc (card ?B)"
+                  proof(rule card_insert_disjoint)
+                    have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)" 
+                      apply (auto simp:image_def)
+                      by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
+                    with vt_s.finite_RAG
+                    show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
+                  next
+                    show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
+                    proof
+                      assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
+                      hence "(Cs cs, Th th) \<in> RAG s" by simp
+                      with True neq_th eq_wq show False
+                        by (auto simp:next_th_def s_RAG_def cs_holding_def)
+                    qed
+                  qed
+                  finally show ?thesis .
+                qed
+              qed
+              moreover note eq_cnp eq_cnv
+              ultimately show ?thesis by simp
+            qed
+          qed
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_set thread prio)
+      assume eq_e: "e = Set thread prio"
+        and is_runing: "thread \<in> runing s"
+      show ?thesis
+      proof -
+        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+        have eq_cncs: "cntCS (e#s) th = cntCS s th"
+          unfolding cntCS_def holdents_test
+          by (simp add:RAG_set_unchanged eq_e)
+        from eq_e have eq_readys: "readys (e#s) = readys s" 
+          by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
+                  auto simp:Let_def)
+        { assume "th \<noteq> thread"
+          with eq_readys eq_e
+          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+            by (simp add:threads.simps)
+          with eq_cnp eq_cnv eq_cncs ih is_runing
+          have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          with is_runing ih have " cntP s th  = cntV s th + cntCS s th" 
+            by (unfold runing_def, auto)
+          moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
+            by (simp add:runing_def)
+          moreover note eq_cnp eq_cnv eq_cncs
+          ultimately have ?thesis by auto
+        } ultimately show ?thesis by blast
+      qed   
+    qed
+  next
+    case vt_nil
+    show ?case 
+      by (unfold cntP_def cntV_def cntCS_def, 
+        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma not_thread_cncs:
+  assumes not_in: "th \<notin> threads s" 
+  shows "cntCS s th = 0"
+proof -
+  from vt not_in show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e th)
+    interpret vt_s: valid_trace s using vt_cons(1)
+       by (unfold_locales, simp)
+    assume vt: "vt s"
+      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
+      and stp: "step s e"
+      and not_in: "th \<notin> threads (e # s)"
+    from stp show ?case
+    proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+        and not_in': "thread \<notin> threads s"
+      have "cntCS (e # s) th = cntCS s th"
+        apply (unfold eq_e cntCS_def holdents_test)
+        by (simp add:RAG_create_unchanged)
+      moreover have "th \<notin> threads s" 
+      proof -
+        from not_in eq_e show ?thesis by simp
+      qed
+      moreover note ih ultimately show ?thesis by auto
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+      and nh: "holdents s thread = {}"
+      have eq_cns: "cntCS (e # s) th = cntCS s th"
+        apply (unfold eq_e cntCS_def holdents_test)
+        by (simp add:RAG_exit_unchanged)
+      show ?thesis
+      proof(cases "th = thread")
+        case True
+        have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
+        with eq_cns show ?thesis by simp
+      next
+        case False
+        with not_in and eq_e
+        have "th \<notin> threads s" by simp
+        from ih[OF this] and eq_cns show ?thesis by simp
+      qed
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      and is_runing: "thread \<in> runing s"
+      from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      hence "cntCS (e # s) th  = cntCS s th "
+        apply (unfold cntCS_def holdents_test eq_e)
+        by (unfold step_RAG_p[OF vtp], auto)
+      moreover have "cntCS s th = 0"
+      proof(rule ih)
+        from not_in eq_e show "th \<notin> threads s" by simp
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+        and is_runing: "thread \<in> runing s"
+        and hold: "holding s thread cs"
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      from assms thread_V vt stp ih 
+      have vtv: "vt (V thread cs#s)" by auto
+      then interpret vt_v: valid_trace "(V thread cs#s)"
+        by (unfold_locales, simp)
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+      from not_in eq_e eq_wq
+      have "\<not> next_th s thread cs th"
+        apply (auto simp:next_th_def)
+      proof -
+        assume ne: "rest \<noteq> []"
+          and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
+        have "?t \<in> set rest"
+        proof(rule someI2)
+          from vt_v.wq_distinct[of cs] and eq_wq
+          show "distinct rest \<and> set rest = set rest"
+            by (metis distinct.simps(2) vt_s.wq_distinct) 
+        next
+          fix x assume "distinct x \<and> set x = set rest" with ne
+          show "hd x \<in> set rest" by (cases x, auto)
+        qed
+        with eq_wq have "?t \<in> set (wq s cs)" by simp
+        from vt_s.wq_threads[OF this] and ni
+        show False
+          using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)` 
+            ni vt_s.wq_threads by blast 
+      qed
+      moreover note neq_th eq_wq
+      ultimately have "cntCS (e # s) th  = cntCS s th"
+        by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
+      moreover have "cntCS s th = 0"
+      proof(rule ih)
+        from not_in eq_e show "th \<notin> threads s" by simp
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (thread_set thread prio)
+      print_facts
+      assume eq_e: "e = Set thread prio"
+        and is_runing: "thread \<in> runing s"
+      from not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] and eq_e
+      show ?thesis 
+        apply (unfold eq_e cntCS_def holdents_test)
+        by (simp add:RAG_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (unfold cntCS_def, 
+        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+  qed
+qed
+
+end
+
+
+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
+
+end
+
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+thm cpreced_initial
+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
+
+context valid_trace
+begin
+
+lemma runing_unique:
+  assumes runing_1: "th1 \<in> runing s"
+  and runing_2: "th2 \<in> runing s"
+  shows "th1 = th2"
+proof -
+  from runing_1 and runing_2 have "cp s th1 = cp s th2"
+    unfolding runing_def
+    apply(simp)
+    done
+  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
+                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
+    (is "Max (?f ` ?A) = Max (?f ` ?B)")
+    unfolding cp_eq_cpreced 
+    unfolding cpreced_def .
+  obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
+  proof -
+    have h1: "finite (?f ` ?A)"
+    proof -
+      have "finite ?A" 
+      proof -
+        have "finite (dependants (wq s) th1)"
+        proof-
+          have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+              apply (auto simp:image_def)
+              by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
+            moreover have "finite \<dots>"
+            proof -
+              from finite_RAG have "finite (RAG s)" .
+              hence "finite ((RAG (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_RAG_def cs_RAG_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependants_def)
+        qed
+        thus ?thesis by simp
+      qed
+      thus ?thesis by auto
+    qed
+    moreover have h2: "(?f ` ?A) \<noteq> {}"
+    proof -
+      have "?A \<noteq> {}" by simp
+      thus ?thesis by simp
+    qed
+    from Max_in [OF h1 h2]
+    have "Max (?f ` ?A) \<in> (?f ` ?A)" .
+    thus ?thesis 
+      thm cpreced_def
+      unfolding cpreced_def[symmetric] 
+      unfolding cp_eq_cpreced[symmetric] 
+      unfolding cpreced_def 
+      using that[intro] by (auto)
+  qed
+  obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
+  proof -
+    have h1: "finite (?f ` ?B)"
+    proof -
+      have "finite ?B" 
+      proof -
+        have "finite (dependants (wq s) th2)"
+        proof-
+          have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+              apply (auto simp:image_def)
+              by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
+            moreover have "finite \<dots>"
+            proof -
+              from finite_RAG have "finite (RAG s)" .
+              hence "finite ((RAG (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_RAG_def cs_RAG_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependants_def)
+        qed
+        thus ?thesis by simp
+      qed
+      thus ?thesis by auto
+    qed
+    moreover have h2: "(?f ` ?B) \<noteq> {}"
+    proof -
+      have "?B \<noteq> {}" by simp
+      thus ?thesis by simp
+    qed
+    from Max_in [OF h1 h2]
+    have "Max (?f ` ?B) \<in> (?f ` ?B)" .
+    thus ?thesis by (auto intro:that)
+  qed
+  from eq_f_th1 eq_f_th2 eq_max 
+  have eq_preced: "preced th1' s = preced th2' s" by auto
+  hence eq_th12: "th1' = th2'"
+  proof (rule preced_unique)
+    from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
+    thus "th1' \<in> threads s"
+    proof
+      assume "th1' \<in> dependants (wq s) th1"
+      hence "(Th th1') \<in> Domain ((RAG s)^+)"
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+        by (auto simp:Domain_def)
+      hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      from dm_RAG_threads[OF this] show ?thesis .
+    next
+      assume "th1' = th1"
+      with runing_1 show ?thesis
+        by (unfold runing_def readys_def, auto)
+    qed
+  next
+    from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
+    thus "th2' \<in> threads s"
+    proof
+      assume "th2' \<in> dependants (wq s) th2"
+      hence "(Th th2') \<in> Domain ((RAG s)^+)"
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+        by (auto simp:Domain_def)
+      hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      from dm_RAG_threads[OF this] show ?thesis .
+    next
+      assume "th2' = th2"
+      with runing_2 show ?thesis
+        by (unfold runing_def readys_def, auto)
+    qed
+  qed
+  from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
+  thus ?thesis
+  proof
+    assume eq_th': "th1' = th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+    thus ?thesis
+    proof
+      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+    next
+      assume "th2' \<in> dependants (wq s) th2"
+      with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
+      hence "(Th th1, Th th2) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      hence "Th th1 \<in> Domain ((RAG s)^+)" 
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+        by (auto simp:Domain_def)
+      hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
+      from RAG_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th1, Cs cs') \<in> RAG s" by simp
+      with runing_1 have "False"
+        apply (unfold runing_def readys_def s_RAG_def)
+        by (auto simp:waiting_eq)
+      thus ?thesis by simp
+    qed
+  next
+    assume th1'_in: "th1' \<in> dependants (wq s) th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+    thus ?thesis 
+    proof
+      assume "th2' = th2"
+      with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
+      hence "(Th th2, Th th1) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      hence "Th th2 \<in> Domain ((RAG s)^+)" 
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+        by (auto simp:Domain_def)
+      hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
+      from RAG_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th2, Cs cs') \<in> RAG s" by simp
+      with runing_2 have "False"
+        apply (unfold runing_def readys_def s_RAG_def)
+        by (auto simp:waiting_eq)
+      thus ?thesis by simp
+    next
+      assume "th2' \<in> dependants (wq s) th2"
+      with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
+      hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      show ?thesis
+      proof(rule dchain_unique[OF h1 _ h2, symmetric])
+        from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
+        from runing_2 show "th2 \<in> readys s" by (simp add:runing_def) 
+      qed
+    qed
+  qed
+qed
+
+
+lemma "card (runing s) \<le> 1"
+apply(subgoal_tac "finite (runing s)")
+prefer 2
+apply (metis finite_nat_set_iff_bounded lessI runing_unique)
+apply(rule ccontr)
+apply(simp)
+apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
+apply(subst (asm) card_le_Suc_iff)
+apply(simp)
+apply(auto)[1]
+apply (metis insertCI runing_unique)
+apply(auto) 
+done
+
+end
+
+
+lemma create_pre:
+  assumes stp: "step s e"
+  and not_in: "th \<notin> threads s"
+  and is_in: "th \<in> threads (e#s)"
+  obtains prio where "e = Create th prio"
+proof -
+  from assms  
+  show ?thesis
+  proof(cases)
+    case (thread_create thread prio)
+    with is_in not_in have "e = Create th prio" by simp
+    from that[OF this] show ?thesis .
+  next
+    case (thread_exit thread)
+    with assms show ?thesis by (auto intro!:that)
+  next
+    case (thread_P thread)
+    with assms show ?thesis by (auto intro!:that)
+  next
+    case (thread_V thread)
+    with assms show ?thesis by (auto intro!:that)
+  next 
+    case (thread_set thread)
+    with assms show ?thesis by (auto intro!:that)
+  qed
+qed
+
+context valid_trace
+begin
+
+lemma cnp_cnv_eq:
+  assumes "th \<notin> threads s"
+  shows "cntP s th = cntV s th"
+  using assms
+  using cnp_cnv_cncs not_thread_cncs by auto
+
+end
+
+
+lemma eq_RAG: 
+  "RAG (wq s) = RAG s"
+by (unfold cs_RAG_def s_RAG_def, auto)
+
+context valid_trace
+begin
+
+lemma count_eq_dependants:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "dependants (wq s) th = {}"
+proof -
+  from cnp_cnv_cncs and eq_pv
+  have "cntCS s th = 0" 
+    by (auto split:if_splits)
+  moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
+  proof -
+    from finite_holding[of th] show ?thesis
+      by (simp add:holdents_test)
+  qed
+  ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
+    by (unfold cntCS_def holdents_test cs_dependants_def, auto)
+  show ?thesis
+  proof(unfold cs_dependants_def)
+    { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
+      then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
+      hence "False"
+      proof(cases)
+        assume "(Th th', Th th) \<in> RAG (wq s)"
+        thus "False" by (auto simp:cs_RAG_def)
+      next
+        fix c
+        assume "(c, Th th) \<in> RAG (wq s)"
+        with h and eq_RAG show "False"
+          by (cases c, auto simp:cs_RAG_def)
+      qed
+    } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
+  qed
+qed
+
+lemma dependants_threads:
+  shows "dependants (wq s) th \<subseteq> threads s"
+proof
+  { fix th th'
+    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
+    have "Th th \<in> Domain (RAG s)"
+    proof -
+      from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
+      hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
+      with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
+      thus ?thesis using eq_RAG by simp
+    qed
+    from dm_RAG_threads[OF this]
+    have "th \<in> threads s" .
+  } note hh = this
+  fix th1 
+  assume "th1 \<in> dependants (wq s) th"
+  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
+    by (unfold cs_dependants_def, simp)
+  from hh [OF this] show "th1 \<in> threads s" .
+qed
+
+lemma finite_threads:
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+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
+
+context valid_trace
+begin
+
+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 le_cp:
+  shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+  show "Prc (priority th s) (last_set th s)
+    \<le> Max (insert (Prc (priority th s) (last_set th s))
+            ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
+    (is "?l \<le> Max (insert ?l ?A)")
+  proof(cases "?A = {}")
+    case False
+    have "finite ?A" (is "finite (?f ` ?B)")
+    proof -
+      have "finite ?B" 
+      proof-
+        have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
+        proof -
+          let ?F = "\<lambda> (x, y). the_th x"
+          have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+            apply (auto simp:image_def)
+            by (rule_tac x = "(Th x, Th th)" in bexI, auto)
+          moreover have "finite \<dots>"
+          proof -
+            from finite_RAG have "finite (RAG s)" .
+            hence "finite ((RAG (wq s))\<^sup>+)"
+              apply (unfold finite_trancl)
+              by (auto simp: s_RAG_def cs_RAG_def wq_def)
+            thus ?thesis by auto
+          qed
+          ultimately show ?thesis by (auto intro:finite_subset)
+        qed
+        thus ?thesis by (simp add:cs_dependants_def)
+      qed
+      thus ?thesis by simp
+    qed
+    from Max_insert [OF this False, of ?l] show ?thesis by auto
+  next
+    case True
+    thus ?thesis by auto
+  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'" by auto
+    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_readys_threads_pre:
+  assumes np: "threads s \<noteq> {}"
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq)
+  show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
+  proof -
+    let ?p = "Max ((\<lambda>th. preced th s) ` threads s)" 
+    let ?f = "(\<lambda>th. preced th s)"
+    have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
+    proof(rule Max_in)
+      from finite_threads show "finite (?f ` threads s)" by simp
+    next
+      from np show "?f ` threads s \<noteq> {}" by simp
+    qed
+    then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
+      by (auto simp:Image_def)
+    from th_chain_to_ready [OF tm_in]
+    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
+    thus ?thesis
+    proof
+      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
+      then obtain th' where th'_in: "th' \<in> readys s" 
+        and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
+      have "cp s th' = ?f tm"
+      proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
+        from dependants_threads finite_threads
+        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))" 
+          by (auto intro:finite_subset)
+      next
+        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+        from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
+        moreover have "p \<le> \<dots>"
+        proof(rule Max_ge)
+          from finite_threads
+          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        next
+          from p_in and th'_in and dependants_threads[of th']
+          show "p \<in> (\<lambda>th. preced th s) ` threads s"
+            by (auto simp:readys_def)
+        qed
+        ultimately show "p \<le> preced tm s" by auto
+      next
+        show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+        proof -
+          from tm_chain
+          have "tm \<in> dependants (wq s) th'"
+            by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
+          thus ?thesis by auto
+        qed
+      qed
+      with tm_max
+      have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+      show ?thesis
+      proof (fold h, rule Max_eqI)
+        fix q 
+        assume "q \<in> cp s ` readys s"
+        then obtain th1 where th1_in: "th1 \<in> readys s"
+          and eq_q: "q = cp s th1" by auto
+        show "q \<le> cp s th'"
+          apply (unfold h eq_q)
+          apply (unfold cp_eq_cpreced cpreced_def)
+          apply (rule Max_mono)
+        proof -
+          from dependants_threads [of th1] th1_in
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq> 
+                 (\<lambda>th. preced th s) ` threads s"
+            by (auto simp:readys_def)
+        next
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
+        next
+          from finite_threads 
+          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        qed
+      next
+        from finite_threads
+        show "finite (cp s ` readys s)" by (auto simp:readys_def)
+      next
+        from th'_in
+        show "cp s th' \<in> cp s ` readys s" by simp
+      qed
+    next
+      assume tm_ready: "tm \<in> readys s"
+      show ?thesis
+      proof(fold tm_max)
+        have cp_eq_p: "cp s tm = preced tm s"
+        proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+          fix y 
+          assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+          show "y \<le> preced tm s"
+          proof -
+            { fix y'
+              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
+              have "y' \<le> preced tm s"
+              proof(unfold tm_max, rule Max_ge)
+                from hy' dependants_threads[of tm]
+                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+              next
+                from finite_threads
+                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+              qed
+            } with hy show ?thesis by auto
+          qed
+        next
+          from dependants_threads[of tm] finite_threads
+          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
+            by (auto intro:finite_subset)
+        next
+          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+            by simp
+        qed 
+        moreover have "Max (cp s ` readys s) = cp s tm"
+        proof(rule Max_eqI)
+          from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
+        next
+          from finite_threads
+          show "finite (cp s ` readys s)" by (auto simp:readys_def)
+        next
+          fix y assume "y \<in> cp s ` readys s"
+          then obtain th1 where th1_readys: "th1 \<in> readys s"
+            and h: "y = cp s th1" by auto
+          show "y \<le> cp s tm"
+            apply(unfold cp_eq_p h)
+            apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
+          proof -
+            from finite_threads
+            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+          next
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
+              by simp
+          next
+            from dependants_threads[of th1] th1_readys
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) 
+                    \<subseteq> (\<lambda>th. preced th s) ` threads s"
+              by (auto simp:readys_def)
+          qed
+        qed
+        ultimately show " Max (cp s ` readys s) = preced tm s" by simp
+      qed 
+    qed
+  qed
+qed
+
+text {* (* ccc *) \noindent
+  Since the current precedence of the threads in ready queue will always be boosted,
+  there must be one inside it has the maximum precedence of the whole system. 
+*}
+lemma max_cp_readys_threads:
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis 
+    by (auto simp:readys_def)
+next
+  case False
+  show ?thesis by (rule max_cp_readys_threads_pre[OF False])
+qed
+
+end
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+  apply (unfold s_holding_def cs_holding_def wq_def, simp)
+  done
+
+lemma f_image_eq:
+  assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
+  shows "f ` A = g ` A"
+proof
+  show "f ` A \<subseteq> g ` A"
+    by(rule image_subsetI, auto intro:h)
+next
+  show "g ` A \<subseteq> f ` A"
+   by (rule image_subsetI, auto intro:h[symmetric])
+qed
+
+
+definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
+  where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
+
+lemma detached_test:
+  shows "detached s th = (Th th \<notin> Field (RAG s))"
+apply(simp add: detached_def Field_def)
+apply(simp add: s_RAG_def)
+apply(simp add: s_holding_abv s_waiting_abv)
+apply(simp add: Domain_iff Range_iff)
+apply(simp add: wq_def)
+apply(auto)
+done
+
+context valid_trace
+begin
+
+lemma detached_intro:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "detached s th"
+proof -
+ from cnp_cnv_cncs
+  have eq_cnt: "cntP s th =
+    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+  hence cncs_zero: "cntCS s th = 0"
+    by (auto simp:eq_pv split:if_splits)
+  with eq_cnt
+  have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
+  thus ?thesis
+  proof
+    assume "th \<notin> threads s"
+    with range_in dm_RAG_threads
+    show ?thesis
+      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
+  next
+    assume "th \<in> readys s"
+    moreover have "Th th \<notin> Range (RAG s)"
+    proof -
+      from card_0_eq [OF finite_holding] and cncs_zero
+      have "holdents s th = {}"
+        by (simp add:cntCS_def)
+      thus ?thesis
+        apply(auto simp:holdents_test)
+        apply(case_tac a)
+        apply(auto simp:holdents_test s_RAG_def)
+        done
+    qed
+    ultimately show ?thesis
+      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
+  qed
+qed
+
+lemma detached_elim:
+  assumes dtc: "detached s th"
+  shows "cntP s th = cntV s th"
+proof -
+  from cnp_cnv_cncs
+  have eq_pv: " cntP s th =
+    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+  have cncs_z: "cntCS s th = 0"
+  proof -
+    from dtc have "holdents s th = {}"
+      unfolding detached_def holdents_test s_RAG_def
+      by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
+    thus ?thesis by (auto simp:cntCS_def)
+  qed
+  show ?thesis
+  proof(cases "th \<in> threads s")
+    case True
+    with dtc 
+    have "th \<in> readys s"
+      by (unfold readys_def detached_def Field_def Domain_def Range_def, 
+           auto simp:waiting_eq s_RAG_def)
+    with cncs_z and eq_pv show ?thesis by simp
+  next
+    case False
+    with cncs_z and eq_pv show ?thesis by simp
+  qed
+qed
+
+lemma detached_eq:
+  shows "(detached s th) = (cntP s th = cntV s th)"
+  by (insert vt, auto intro:detached_intro detached_elim)
+
+end
+
+text {* 
+  The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived
+  from the concise and miniature model of PIP given in PrioGDef.thy.
+*}
+
+lemma eq_dependants: "dependants (wq s) = dependants s"
+  by (simp add: s_dependants_abv wq_def)
+
+lemma next_th_unique: 
+  assumes nt1: "next_th s th cs th1"
+  and nt2: "next_th s th cs th2"
+  shows "th1 = th2"
+using assms by (unfold next_th_def, auto)
+
+lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
+  apply (induct s, simp)
+proof -
+  fix a s
+  assume ih: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
+    and eq_as: "a # s \<noteq> []"
+  show "last_set th (a # s) < length (a # s)"
+  proof(cases "s \<noteq> []")
+    case False
+    from False show ?thesis
+      by (cases a, auto simp:last_set.simps)
+  next
+    case True
+    from ih [OF True] show ?thesis
+      by (cases a, auto simp:last_set.simps)
+  qed
+qed
+
+lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
+  by (induct s, auto simp:threads.simps)
+
+lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
+  apply (drule_tac th_in_ne)
+  by (unfold preced_def, auto intro: birth_time_lt)
+
+lemma inj_the_preced: 
+  "inj_on (the_preced s) (threads s)"
+  by (metis inj_onI preced_unique the_preced_def)
+
+lemma tRAG_alt_def: 
+  "tRAG s = {(Th th1, Th th2) | th1 th2. 
+                  \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
+ by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
+
+lemma tRAG_Field:
+  "Field (tRAG s) \<subseteq> Field (RAG s)"
+  by (unfold tRAG_alt_def Field_def, auto)
+
+lemma tRAG_ancestorsE:
+  assumes "x \<in> ancestors (tRAG s) u"
+  obtains th where "x = Th th"
+proof -
+  from assms have "(u, x) \<in> (tRAG s)^+" 
+      by (unfold ancestors_def, auto)
+  from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
+  then obtain th where "x = Th th"
+    by (unfold tRAG_alt_def, auto)
+  from that[OF this] show ?thesis .
+qed
+
+lemma tRAG_mono:
+  assumes "RAG s' \<subseteq> RAG s"
+  shows "tRAG s' \<subseteq> tRAG s"
+  using assms 
+  by (unfold tRAG_alt_def, auto)
+
+lemma holding_next_thI:
+  assumes "holding s th cs"
+  and "length (wq s cs) > 1"
+  obtains th' where "next_th s th cs th'"
+proof -
+  from assms(1)[folded eq_holding, unfolded cs_holding_def]
+  have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
+  then obtain rest where h1: "wq s cs = th#rest" 
+    by (cases "wq s cs", auto)
+  with assms(2) have h2: "rest \<noteq> []" by auto
+  let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+  have "next_th s th cs ?th'" using  h1(1) h2 
+    by (unfold next_th_def, auto)
+  from that[OF this] show ?thesis .
+qed
+
+lemma RAG_tRAG_transfer:
+  assumes "vt s'"
+  assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+  and "(Cs cs, Th th'') \<in> RAG s'"
+  shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
+proof -
+  interpret vt_s': valid_trace "s'" using assms(1)
+    by (unfold_locales, simp)
+  interpret rtree: rtree "RAG s'"
+  proof
+  show "single_valued (RAG s')"
+  apply (intro_locales)
+    by (unfold single_valued_def, 
+        auto intro:vt_s'.unique_RAG)
+
+  show "acyclic (RAG s')"
+     by (rule vt_s'.acyclic_RAG)
+  qed
+  { fix n1 n2
+    assume "(n1, n2) \<in> ?L"
+    from this[unfolded tRAG_alt_def]
+    obtain th1 th2 cs' where 
+      h: "n1 = Th th1" "n2 = Th th2" 
+         "(Th th1, Cs cs') \<in> RAG s"
+         "(Cs cs', Th th2) \<in> RAG s" by auto
+    from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
+    from h(3) and assms(2) 
+    have "(Th th1, Cs cs') = (Th th, Cs cs) \<or> 
+          (Th th1, Cs cs') \<in> RAG s'" by auto
+    hence "(n1, n2) \<in> ?R"
+    proof
+      assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
+      hence eq_th1: "th1 = th" by simp
+      moreover have "th2 = th''"
+      proof -
+        from h1 have "cs' = cs" by simp
+        from assms(3) cs_in[unfolded this] rtree.sgv
+        show ?thesis
+          by (unfold single_valued_def, auto)
+      qed
+      ultimately show ?thesis using h(1,2) by auto
+    next
+      assume "(Th th1, Cs cs') \<in> RAG s'"
+      with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
+        by (unfold tRAG_alt_def, auto)
+      from this[folded h(1, 2)] show ?thesis by auto
+    qed
+  } moreover {
+    fix n1 n2
+    assume "(n1, n2) \<in> ?R"
+    hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
+    hence "(n1, n2) \<in> ?L" 
+    proof
+      assume "(n1, n2) \<in> tRAG s'"
+      moreover have "... \<subseteq> ?L"
+      proof(rule tRAG_mono)
+        show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
+      qed
+      ultimately show ?thesis by auto
+    next
+      assume eq_n: "(n1, n2) = (Th th, Th th'')"
+      from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
+      moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
+      ultimately show ?thesis 
+        by (unfold eq_n tRAG_alt_def, auto)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+context valid_trace
+begin
+
+lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
+
+end
+
+lemma cp_alt_def:
+  "cp s th =  
+           Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
+proof -
+  have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
+        Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})" 
+          (is "Max (_ ` ?L) = Max (_ ` ?R)")
+  proof -
+    have "?L = ?R" 
+    by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
+    thus ?thesis by simp
+  qed
+  thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
+qed
+
+lemma cp_gen_alt_def:
+  "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
+    by (auto simp:cp_gen_def)
+
+lemma tRAG_nodeE:
+  assumes "(n1, n2) \<in> tRAG s"
+  obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
+  using assms
+  by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
+
+lemma subtree_nodeE:
+  assumes "n \<in> subtree (tRAG s) (Th th)"
+  obtains th1 where "n = Th th1"
+proof -
+  show ?thesis
+  proof(rule subtreeE[OF assms])
+    assume "n = Th th"
+    from that[OF this] show ?thesis .
+  next
+    assume "Th th \<in> ancestors (tRAG s) n"
+    hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+    hence "\<exists> th1. n = Th th1"
+    proof(induct)
+      case (base y)
+      from tRAG_nodeE[OF this] show ?case by metis
+    next
+      case (step y z)
+      thus ?case by auto
+    qed
+    with that show ?thesis by auto
+  qed
+qed
+
+lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
+proof -
+  have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*" 
+    by (rule rtrancl_mono, auto simp:RAG_split)
+  also have "... \<subseteq> ((RAG s)^*)^*"
+    by (rule rtrancl_mono, auto)
+  also have "... = (RAG s)^*" by simp
+  finally show ?thesis by (unfold tRAG_def, simp)
+qed
+
+lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
+proof -
+  { fix a
+    assume "a \<in> subtree (tRAG s) x"
+    hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
+    with tRAG_star_RAG[of s]
+    have "(a, x) \<in> (RAG s)^*" by auto
+    hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
+  } thus ?thesis by auto
+qed
+
+lemma tRAG_trancl_eq:
+   "{th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
+    {th'. (Th th', Th th)  \<in> (RAG s)^+}"
+   (is "?L = ?R")
+proof -
+  { fix th'
+    assume "th' \<in> ?L"
+    hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
+    from tranclD[OF this]
+    obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
+    from tRAG_subtree_RAG[of s] and this(2)
+    have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG) 
+    moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto 
+    ultimately have "th' \<in> ?R"  by auto 
+  } moreover 
+  { fix th'
+    assume "th' \<in> ?R"
+    hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
+    from plus_rpath[OF this]
+    obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
+    hence "(Th th', Th th) \<in> (tRAG s)^+"
+    proof(induct xs arbitrary:th' th rule:length_induct)
+      case (1 xs th' th)
+      then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
+      show ?case
+      proof(cases "xs1")
+        case Nil
+        from 1(2)[unfolded Cons1 Nil]
+        have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
+        hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
+        then obtain cs where "x1 = Cs cs" 
+              by (unfold s_RAG_def, auto)
+        from rpath_nnl_lastE[OF rp[unfolded this]]
+        show ?thesis by auto
+      next
+        case (Cons x2 xs2)
+        from 1(2)[unfolded Cons1[unfolded this]]
+        have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
+        from rpath_edges_on[OF this]
+        have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
+        have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+            by (simp add: edges_on_unfold)
+        with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
+        then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
+        have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+            by (simp add: edges_on_unfold)
+        from this eds
+        have rg2: "(x1, x2) \<in> RAG s" by auto
+        from this[unfolded eq_x1] 
+        obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
+        from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
+        have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
+        from rp have "rpath (RAG s) x2 xs2 (Th th)"
+           by  (elim rpath_ConsE, simp)
+        from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
+        show ?thesis
+        proof(cases "xs2 = []")
+          case True
+          from rpath_nilE[OF rp'[unfolded this]]
+          have "th1 = th" by auto
+          from rt1[unfolded this] show ?thesis by auto
+        next
+          case False
+          from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
+          have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
+          with rt1 show ?thesis by auto
+        qed
+      qed
+    qed
+    hence "th' \<in> ?L" by auto
+  } ultimately show ?thesis by blast
+qed
+
+lemma tRAG_trancl_eq_Th:
+   "{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
+    {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}"
+    using tRAG_trancl_eq by auto
+
+lemma dependants_alt_def:
+  "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
+  by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
+  
+context valid_trace
+begin
+
+lemma count_eq_tRAG_plus:
+  assumes "cntP s th = cntV s th"
+  shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+  using assms count_eq_dependants dependants_alt_def eq_dependants by auto 
+
+lemma count_eq_RAG_plus:
+  assumes "cntP s th = cntV s th"
+  shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+  using assms count_eq_dependants cs_dependants_def eq_RAG by auto
+
+lemma count_eq_RAG_plus_Th:
+  assumes "cntP s th = cntV s th"
+  shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+  using count_eq_RAG_plus[OF assms] by auto
+
+lemma count_eq_tRAG_plus_Th:
+  assumes "cntP s th = cntV s th"
+  shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+   using count_eq_tRAG_plus[OF assms] by auto
+
+end
+
+lemma tRAG_subtree_eq: 
+   "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th'  \<in> (subtree (RAG s) (Th th))}"
+   (is "?L = ?R")
+proof -
+  { fix n 
+    assume h: "n \<in> ?L"
+    hence "n \<in> ?R"
+    by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) 
+  } moreover {
+    fix n
+    assume "n \<in> ?R"
+    then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
+      by (auto simp:subtree_def)
+    from rtranclD[OF this(2)]
+    have "n \<in> ?L"
+    proof
+      assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
+      with h have "n \<in> {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}" by auto
+      thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
+    qed (insert h, auto simp:subtree_def)
+  } ultimately show ?thesis by auto
+qed
+
+lemma threads_set_eq: 
+   "the_thread ` (subtree (tRAG s) (Th th)) = 
+                  {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
+   by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
+
+lemma cp_alt_def1: 
+  "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
+proof -
+  have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
+       ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
+       by auto
+  thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
+qed
+
+lemma cp_gen_def_cond: 
+  assumes "x = Th th"
+  shows "cp s th = cp_gen s (Th th)"
+by (unfold cp_alt_def1 cp_gen_def, simp)
+
+lemma cp_gen_over_set:
+  assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
+  shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
+proof(rule f_image_eq)
+  fix a
+  assume "a \<in> A"
+  from assms[rule_format, OF this]
+  obtain th where eq_a: "a = Th th" by auto
+  show "cp_gen s a = (cp s \<circ> the_thread) a"
+    by  (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
+qed
+
+
+context valid_trace
+begin
+
+lemma RAG_threads:
+  assumes "(Th th) \<in> Field (RAG s)"
+  shows "th \<in> threads s"
+  using assms
+  by (metis Field_def UnE dm_RAG_threads range_in vt)
+
+lemma subtree_tRAG_thread:
+  assumes "th \<in> threads s"
+  shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
+proof -
+  have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+    by (unfold tRAG_subtree_eq, simp)
+  also have "... \<subseteq> ?R"
+  proof
+    fix x
+    assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+    then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
+    from this(2)
+    show "x \<in> ?R"
+    proof(cases rule:subtreeE)
+      case 1
+      thus ?thesis by (simp add: assms h(1)) 
+    next
+      case 2
+      thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) 
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma readys_root:
+  assumes "th \<in> readys s"
+  shows "root (RAG s) (Th th)"
+proof -
+  { fix x
+    assume "x \<in> ancestors (RAG s) (Th th)"
+    hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+    from tranclD[OF this]
+    obtain z where "(Th th, z) \<in> RAG s" by auto
+    with assms(1) have False
+         apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
+         by (fold wq_def, blast)
+  } thus ?thesis by (unfold root_def, auto)
+qed
+
+lemma readys_in_no_subtree:
+  assumes "th \<in> readys s"
+  and "th' \<noteq> th"
+  shows "Th th \<notin> subtree (RAG s) (Th th')" 
+proof
+   assume "Th th \<in> subtree (RAG s) (Th th')"
+   thus False
+   proof(cases rule:subtreeE)
+      case 1
+      with assms show ?thesis by auto
+   next
+      case 2
+      with readys_root[OF assms(1)]
+      show ?thesis by (auto simp:root_def)
+   qed
+qed
+
+lemma not_in_thread_isolated:
+  assumes "th \<notin> threads s"
+  shows "(Th th) \<notin> Field (RAG s)"
+proof
+  assume "(Th th) \<in> Field (RAG s)"
+  with dm_RAG_threads and range_in assms
+  show False by (unfold Field_def, blast)
+qed
+
+lemma wf_RAG: "wf (RAG s)"
+proof(rule finite_acyclic_wf)
+  from finite_RAG show "finite (RAG s)" .
+next
+  from acyclic_RAG show "acyclic (RAG s)" .
+qed
+
+lemma sgv_wRAG: "single_valued (wRAG s)"
+  using waiting_unique
+  by (unfold single_valued_def wRAG_def, auto)
+
+lemma sgv_hRAG: "single_valued (hRAG s)"
+  using holding_unique 
+  by (unfold single_valued_def hRAG_def, auto)
+
+lemma sgv_tRAG: "single_valued (tRAG s)"
+  by (unfold tRAG_def, rule single_valued_relcomp, 
+              insert sgv_wRAG sgv_hRAG, auto)
+
+lemma acyclic_tRAG: "acyclic (tRAG s)"
+proof(unfold tRAG_def, rule acyclic_compose)
+  show "acyclic (RAG s)" using acyclic_RAG .
+next
+  show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+next
+  show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+qed
+
+lemma sgv_RAG: "single_valued (RAG s)"
+  using unique_RAG by (auto simp:single_valued_def)
+
+lemma rtree_RAG: "rtree (RAG s)"
+  using sgv_RAG acyclic_RAG
+  by (unfold rtree_def rtree_axioms_def sgv_def, auto)
+
+end
+
+sublocale valid_trace < rtree_RAG: rtree "RAG s"
+proof
+  show "single_valued (RAG s)"
+  apply (intro_locales)
+    by (unfold single_valued_def, 
+        auto intro:unique_RAG)
+
+  show "acyclic (RAG s)"
+     by (rule acyclic_RAG)
+qed
+
+sublocale valid_trace < rtree_s: rtree "tRAG s"
+proof(unfold_locales)
+  from sgv_tRAG show "single_valued (tRAG s)" .
+next
+  from acyclic_tRAG show "acyclic (tRAG s)" .
+qed
+
+sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
+proof -
+  show "fsubtree (RAG s)"
+  proof(intro_locales)
+    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
+  next
+    show "fsubtree_axioms (RAG s)"
+    proof(unfold fsubtree_axioms_def)
+      from wf_RAG show "wf (RAG s)" .
+    qed
+  qed
+qed
+
+sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
+proof -
+  have "fsubtree (tRAG s)"
+  proof -
+    have "fbranch (tRAG s)"
+    proof(unfold tRAG_def, rule fbranch_compose)
+        show "fbranch (wRAG s)"
+        proof(rule finite_fbranchI)
+           from finite_RAG show "finite (wRAG s)"
+           by (unfold RAG_split, auto)
+        qed
+    next
+        show "fbranch (hRAG s)"
+        proof(rule finite_fbranchI)
+           from finite_RAG 
+           show "finite (hRAG s)" by (unfold RAG_split, auto)
+        qed
+    qed
+    moreover have "wf (tRAG s)"
+    proof(rule wf_subset)
+      show "wf (RAG s O RAG s)" using wf_RAG
+        by (fold wf_comp_self, simp)
+    next
+      show "tRAG s \<subseteq> (RAG s O RAG s)"
+        by (unfold tRAG_alt_def, auto)
+    qed
+    ultimately show ?thesis
+      by (unfold fsubtree_def fsubtree_axioms_def,auto)
+  qed
+  from this[folded tRAG_def] show "fsubtree (tRAG s)" .
+qed
+
+lemma Max_UNION: 
+  assumes "finite A"
+  and "A \<noteq> {}"
+  and "\<forall> M \<in> f ` A. finite M"
+  and "\<forall> M \<in> f ` A. M \<noteq> {}"
+  shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
+  using assms[simp]
+proof -
+  have "?L = Max (\<Union>(f ` A))"
+    by (fold Union_image_eq, simp)
+  also have "... = ?R"
+    by (subst Max_Union, simp+)
+  finally show ?thesis .
+qed
+
+lemma max_Max_eq:
+  assumes "finite A"
+    and "A \<noteq> {}"
+    and "x = y"
+  shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
+proof -
+  have "?R = Max (insert y A)" by simp
+  also from assms have "... = ?L"
+      by (subst Max.insert, simp+)
+  finally show ?thesis by simp
+qed
+
+context valid_trace
+begin
+
+(* ddd *)
+lemma cp_gen_rec:
+  assumes "x = Th th"
+  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
+proof(cases "children (tRAG s) x = {}")
+  case True
+  show ?thesis
+    by (unfold True cp_gen_def subtree_children, simp add:assms)
+next
+  case False
+  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
+  note fsbttRAGs.finite_subtree[simp]
+  have [simp]: "finite (children (tRAG s) x)"
+     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 
+            rule children_subtree)
+  { fix r x
+    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
+  } note this[simp]
+  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
+  proof -
+    from False obtain q where "q \<in> children (tRAG s) x" by blast
+    moreover have "subtree (tRAG s) q \<noteq> {}" by simp
+    ultimately show ?thesis by blast
+  qed
+  have h: "Max ((the_preced s \<circ> the_thread) `
+                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
+        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
+                     (is "?L = ?R")
+  proof -
+    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
+    let "Max (_ \<union> (?h ` ?B))" = ?R
+    let ?L1 = "?f ` \<Union>(?g ` ?B)"
+    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
+    proof -
+      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
+      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
+      finally have "Max ?L1 = Max ..." by simp
+      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
+        by (subst Max_UNION, simp+)
+      also have "... = Max (cp_gen s ` children (tRAG s) x)"
+          by (unfold image_comp cp_gen_alt_def, simp)
+      finally show ?thesis .
+    qed
+    show ?thesis
+    proof -
+      have "?L = Max (?f ` ?A \<union> ?L1)" by simp
+      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
+            by (subst Max_Un, simp+)
+      also have "... = max (?f x) (Max (?h ` ?B))"
+        by (unfold eq_Max_L1, simp)
+      also have "... =?R"
+        by (rule max_Max_eq, (simp)+, unfold assms, simp)
+      finally show ?thesis .
+    qed
+  qed  thus ?thesis 
+          by (fold h subtree_children, unfold cp_gen_def, simp) 
+qed
+
+lemma cp_rec:
+  "cp s th = Max ({the_preced s th} \<union> 
+                     (cp s o the_thread) ` children (tRAG s) (Th th))"
+proof -
+  have "Th th = Th th" by simp
+  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
+  show ?thesis 
+  proof -
+    have "cp_gen s ` children (tRAG s) (Th th) = 
+                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
+    proof(rule cp_gen_over_set)
+      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
+        by (unfold tRAG_alt_def, auto simp:children_def)
+    qed
+    thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+  qed
+qed
+
+end
+
+(* keep *)
+lemma next_th_holding:
+  assumes vt: "vt s"
+  and nxt: "next_th s th cs th'"
+  shows "holding (wq s) th cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  thus ?thesis
+    by (unfold cs_holding_def, auto)
+qed
+
+context valid_trace
+begin
+
+lemma next_th_waiting:
+  assumes nxt: "next_th s th cs th'"
+  shows "waiting (wq s) th' cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  from wq_distinct[of cs, unfolded h]
+  have dst: "distinct (th # rest)" .
+  have in_rest: "th' \<in> set rest"
+  proof(unfold h, rule someI2)
+    show "distinct rest \<and> set rest = set rest" using dst by auto
+  next
+    fix x assume "distinct x \<and> set x = set rest"
+    with h(2)
+    show "hd x \<in> set (rest)" by (cases x, auto)
+  qed
+  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
+  moreover have "th' \<noteq> hd (wq s cs)"
+    by (unfold h(1), insert in_rest dst, auto)
+  ultimately show ?thesis by (auto simp:cs_waiting_def)
+qed
+
+lemma next_th_RAG:
+  assumes nxt: "next_th (s::event list) th cs th'"
+  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
+  using vt assms next_th_holding next_th_waiting
+  by (unfold s_RAG_def, simp)
+
+end
+
+-- {* A useless definition *}
+definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+
+lemma "wq (V th cs # s) cs1 = ttt"
+  apply (unfold wq_def, auto simp:Let_def)
+
+end
+




