# HG changeset patch
# User zhangx
# Date 1454331399 -28800
# Node ID 3d2b59f15f2687706e5889795c0e7a5ee2538cae
# Parent  f7b33c633b96ede415a7059c99555cba75a8caad
Reorganizing PIPBasics.thy

diff -r f7b33c633b96 -r 3d2b59f15f26 Moment.thy
--- a/Moment.thy	Sun Jan 31 18:15:13 2016 +0800
+++ b/Moment.thy	Mon Feb 01 20:56:39 2016 +0800
@@ -82,6 +82,12 @@
   using assms 
   by (simp add:moment_def rev_take, 
       metis Suc_diff_le diff_Suc_Suc drop_Suc tl_drop)
+
+lemma moment_Suc_hd:
+  assumes "Suc i \<le> length s"
+  shows "hd (moment (Suc i) s) = s!(length s - Suc i)"
+  by (simp add:moment_def rev_take, 
+      subst hd_drop_conv_nth, insert assms, auto)
   
 lemma moment_plus:
   assumes "Suc i \<le> length s"
diff -r f7b33c633b96 -r 3d2b59f15f26 PIPBasics.thy
--- a/PIPBasics.thy	Sun Jan 31 18:15:13 2016 +0800
+++ b/PIPBasics.thy	Mon Feb 01 20:56:39 2016 +0800
@@ -86,6 +86,12 @@
   finally show ?thesis by simp
 qed
 
+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
+
 section {* Lemmas do not depend on trace validity *}
 
 lemma birth_time_lt:  
@@ -206,6 +212,67 @@
   with eq_wq that show ?thesis by metis
 qed
 
+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) 
+
+
+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
+
+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
+
 (* ccc *)
 
 section {* Locales used to investigate the execution of PIP *}
@@ -479,9 +546,29 @@
 locale valid_moment = valid_trace + 
   fixes i :: nat
 
-sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
+sublocale valid_moment < vat_moment!: valid_trace "(moment i s)"
   by (unfold_locales, insert vt_moment, auto)
 
+locale valid_moment_e = valid_moment +
+  assumes less_i: "i < length s"
+begin
+  definition "next_e  = hd (moment (Suc i) s)"
+
+  lemma trace_e: 
+    "moment (Suc i) s = next_e#moment i s"
+   proof -
+    from less_i have "Suc i \<le> length s" by auto
+    from moment_plus[OF this, folded next_e_def]
+    show ?thesis .
+   qed
+
+end
+
+sublocale valid_moment_e < vat_moment_e!: valid_trace_e "moment i s" "next_e"
+  using vt_moment[of "Suc i", unfolded trace_e]
+  by (unfold_locales, simp)
+
+section {* Distinctiveness of waiting queues *}
 
 context valid_trace_create
 begin
@@ -511,7 +598,7 @@
   using assms by simp
 end
 
-context valid_trace_p (* ccc *)
+context valid_trace_p 
 begin
 
 lemma wq_neq_simp [simp]:
@@ -527,31 +614,6 @@
   show ?thesis by (cases, simp)
 qed
 
-lemma ready_th_s: "th \<in> readys s"
-  using runing_th_s
-  by (unfold runing_def, auto)
-
-lemma live_th_s: "th \<in> threads s"
-  using readys_threads ready_th_s by auto
-
-lemma live_th_es: "th \<in> threads (e#s)"
-  using live_th_s 
-  by (unfold is_p, simp)
-
-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
@@ -595,27 +657,6 @@
     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 -
@@ -665,23 +706,6 @@
 context valid_trace
 begin
 
-lemma actor_inv: (* ccc *)
-  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)
@@ -715,47 +739,12 @@
 
 end
 
+section {* Waiting queues and threads *}
+
 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: (* ccc *)
+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)"
@@ -788,248 +777,173 @@
   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)
 
+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)
+
 end
 
+lemma (in valid_trace_create)
+  th_not_in_threads: "th \<notin> threads s"
+proof -
+  from pip_e[unfolded is_create]
+  show ?thesis by (cases, simp)
+qed
+
+lemma (in valid_trace_create)
+  threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
+  by (unfold is_create, simp)
+
+lemma (in valid_trace_exit)
+  threads_es [simp]: "threads (e#s) = threads s - {th}"
+  by (unfold is_exit, simp)
+
+lemma (in valid_trace_p)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_p, simp)
+
+lemma (in valid_trace_v)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_v, simp)
+
+lemma (in valid_trace_v)
+  th_not_in_rest[simp]: "th \<notin> set rest"
+proof
+  assume otherwise: "th \<in> set rest"
+  have "distinct (wq s cs)" by (simp add: wq_distinct)
+  from this[unfolded wq_s_cs] and otherwise
+  show False by auto
+qed
+
+lemma (in valid_trace_v) distinct_rest: "distinct rest"
+  by (simp add: distinct_tl rest_def wq_distinct)
+
+lemma (in valid_trace_v)
+  set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
+proof(unfold wq_es_cs wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume "distinct x \<and> set x = set rest"
+  thus "set x = set (wq s cs) - {th}" 
+      by (unfold wq_s_cs, simp)
+qed
+
+lemma (in valid_trace_exit)
+  th_not_in_wq: "th \<notin> set (wq s cs)"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis
+  by (cases, unfold holdents_def s_holding_def, fold wq_def, 
+             auto elim!:runing_wqE)
+qed
+
+lemma (in valid_trace) wq_threads: 
+  assumes "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+  using assms
+proof(induct rule:ind)
+  case (Nil)
+  thus ?case by (auto simp:wq_def)
+next
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case
+  proof(cases e)
+    case (Create th' prio')
+    interpret vt: valid_trace_create s e th' prio'
+      using Create by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems by auto
+  next
+    case (Exit th')
+    interpret vt: valid_trace_exit s e th'
+      using Exit by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto 
+  next
+    case (P th' cs')
+    interpret vt: valid_trace_p s e th' cs'
+      using P by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems readys_threads 
+        runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv 
+        by fastforce 
+  next
+    case (V th' cs')
+    interpret vt: valid_trace_v s e th' cs'
+      using V by (unfold_locales, simp)
+    show ?thesis using Cons
+      using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
+  next
+    case (Set th' prio)
+    interpret vt: valid_trace_set s e th' prio
+      using Set by (unfold_locales, simp)
+    show ?thesis using Cons.hyps(2) Cons.prems vt.is_set 
+        by (auto simp:wq_def Let_def)
+  qed
+qed 
+
+section {* RAG and threads *}
 
 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"
+lemma  dm_RAG_threads:
+  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+  shows "th \<in> threads s"
 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
+  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
 
-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
+lemma rg_RAG_threads: 
+  assumes "(Th th) \<in> Range (RAG s)"
+  shows "th \<in> threads s"
+  using assms
+  by (unfold s_RAG_def cs_waiting_def cs_holding_def, 
+       auto intro:wq_threads)
+
+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 rg_RAG_threads)
 
 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
+section {* The change of @{term RAG} *}
 
 text {*
   The following three lemmas show that @{text "RAG"} does not change
@@ -1037,36 +951,18 @@
   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 (in valid_trace_set)
-   RAG_unchanged: "(RAG (e # s)) = RAG s"
-   by (unfold is_set RAG_set_unchanged, simp)
-
-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 (in valid_trace_create)
-   RAG_unchanged: "(RAG (e # s)) = RAG s"
-   by (unfold is_create RAG_create_unchanged, simp)
-
-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)
-
-lemma (in valid_trace_exit)
-   RAG_unchanged: "(RAG (e # s)) = RAG s"
-   by (unfold is_exit RAG_exit_unchanged, simp)
+lemma (in valid_trace_set) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s"
+   by (unfold is_set s_RAG_def s_waiting_def wq_def, simp add:Let_def)
+
+lemma (in valid_trace_create) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s"
+ by (unfold is_create s_RAG_def s_waiting_def wq_def, simp add:Let_def)
+
+lemma (in valid_trace_exit) RAG_unchanged[simp]: "(RAG (e # s)) = RAG s"
+  by (unfold is_exit s_RAG_def s_waiting_def wq_def, simp add:Let_def)
 
 context valid_trace_v
 begin
 
-lemma distinct_rest: "distinct rest"
-  by (simp add: distinct_tl rest_def wq_distinct)
-
 lemma holding_cs_eq_th:
   assumes "holding s t cs"
   shows "t = th"
@@ -1084,14 +980,20 @@
   by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
   
 lemma set_wq': "set wq' = set rest"
-  by (metis (mono_tags, lifting) distinct_rest rest_def 
-      some_eq_ex wq'_def)
+  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 set_wq' by simp
 
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis by (cases, simp)
+qed
+
 lemma neq_t_th: 
   assumes "waiting (e#s) t c"
   shows "t \<noteq> th"
@@ -1114,7 +1016,7 @@
         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
+    with runing_th_s[folded otherwise] show ?thesis by auto 
   qed
 qed
 
@@ -1367,18 +1269,6 @@
 
 end 
 
-lemma rel_eqI:
-  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
-  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
-  shows "A = B"
-  using assms by auto
-
-lemma in_RAG_E:
-  assumes "(n1, n2) \<in> RAG (s::state)"
-  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
-      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
-  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
-  by auto
   
 context valid_trace_v
 begin
@@ -1565,183 +1455,29 @@
  qed
 qed
 
-end
-
-lemma step_RAG_v: 
-assumes vt:
-  "vt (V th cs#s)"
-shows "
-  RAG (V th cs # s) =
-  RAG s - {(Cs cs, Th th)} -
-  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-  {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
-proof -
-  interpret vt_v: valid_trace_v s "V th cs"
-    using assms step_back_vt by (unfold_locales, auto) 
-  show ?thesis using vt_v.RAG_es .
-qed
-
-lemma (in valid_trace_create)
-  th_not_in_threads: "th \<notin> threads s"
-proof -
-  from pip_e[unfolded is_create]
-  show ?thesis by (cases, simp)
-qed
-
-lemma (in valid_trace_create)
-  threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
-  by (unfold is_create, simp)
-
-lemma (in valid_trace_exit)
-  threads_es [simp]: "threads (e#s) = threads s - {th}"
-  by (unfold is_exit, simp)
-
-lemma (in valid_trace_p)
-  threads_es [simp]: "threads (e#s) = threads s"
-  by (unfold is_p, simp)
-
-lemma (in valid_trace_v)
-  threads_es [simp]: "threads (e#s) = threads s"
-  by (unfold is_v, simp)
-
-lemma (in valid_trace_v)
-  th_not_in_rest[simp]: "th \<notin> set rest"
-proof
-  assume otherwise: "th \<in> set rest"
-  have "distinct (wq s cs)" by (simp add: wq_distinct)
-  from this[unfolded wq_s_cs] and otherwise
-  show False by auto
-qed
-
-lemma (in valid_trace_v)
-  set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
-proof(unfold wq_es_cs wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest)
+lemma 
+  finite_RAG_kept:
+  assumes "finite (RAG s)"
+  shows "finite (RAG (e#s))"
+proof(cases "rest = []")
+  case True
+  interpret vt: valid_trace_v_e using True
+    by (unfold_locales, simp)
+  show ?thesis using assms
+    by  (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
 next
-  fix x
-  assume "distinct x \<and> set x = set rest"
-  thus "set x = set (wq s cs) - {th}" 
-      by (unfold wq_s_cs, simp)
-qed
-
-lemma (in valid_trace_exit)
-  th_not_in_wq: "th \<notin> set (wq s cs)"
-proof -
-  from pip_e[unfolded is_exit]
-  show ?thesis
-  by (cases, unfold holdents_def s_holding_def, fold wq_def, 
-             auto elim!:runing_wqE)
+  case False
+  interpret vt: valid_trace_v_n using False
+    by (unfold_locales, simp)
+  show ?thesis using assms
+    by  (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
 qed
 
-lemma (in valid_trace) wq_threads: 
-  assumes "th \<in> set (wq s cs)"
-  shows "th \<in> threads s"
-  using assms
-proof(induct rule:ind)
-  case (Nil)
-  thus ?case by (auto simp:wq_def)
-next
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case
-  proof(cases e)
-    case (Create th' prio')
-    interpret vt: valid_trace_create s e th' prio'
-      using Create by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems by auto
-  next
-    case (Exit th')
-    interpret vt: valid_trace_exit s e th'
-      using Exit by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto 
-  next
-    case (P th' cs')
-    interpret vt: valid_trace_p s e th' cs'
-      using P by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems readys_threads 
-        runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv 
-        by fastforce 
-  next
-    case (V th' cs')
-    interpret vt: valid_trace_v s e th' cs'
-      using V by (unfold_locales, simp)
-    show ?thesis using Cons
-      using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
-  next
-    case (Set th' prio)
-    interpret vt: valid_trace_set s e th' prio
-      using Set by (unfold_locales, simp)
-    show ?thesis using Cons.hyps(2) Cons.prems vt.is_set 
-        by (auto simp:wq_def Let_def)
-  qed
-qed 
-
-context valid_trace
-begin
-
-lemma  dm_RAG_threads:
-  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
-  shows "th \<in> threads s"
-proof -
-  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
-  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
-  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
-  hence "th \<in> set (wq s cs)"
-    by (unfold s_RAG_def, auto simp:cs_waiting_def)
-  from wq_threads [OF this] show ?thesis .
-qed
-
-lemma rg_RAG_threads: 
-  assumes "(Th th) \<in> Range (RAG s)"
-  shows "th \<in> threads s"
-  using assms
-  by (unfold s_RAG_def cs_waiting_def cs_holding_def, 
-       auto intro:wq_threads)
-
-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 rg_RAG_threads)
-
 end
 
-lemma (in valid_trace_v)
-  preced_es [simp]: "preced th (e#s) = preced th s"
-  by (unfold is_v preced_def, simp)
-
-lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
-proof
-  fix th'
-  show "the_preced (V th cs # s) th' = the_preced s th'"
-    by (unfold the_preced_def preced_def, simp)
-qed
-
-lemma (in valid_trace_v)
-  the_preced_es: "the_preced (e#s) = the_preced s"
-  by (unfold is_v preced_def, simp)
-
 context valid_trace_p
 begin
 
-lemma not_holding_s_th_cs: "\<not> holding s th cs"
-proof
-  assume otherwise: "holding s th cs"
-  from pip_e[unfolded is_p]
-  show False
-  proof(cases)
-    case (thread_P)
-    moreover have "(Cs cs, Th th) \<in> RAG s"
-      using otherwise cs_holding_def 
-            holding_eq th_not_in_wq by auto
-    ultimately show ?thesis by auto
-  qed
-qed
-
 lemma waiting_kept:
   assumes "waiting s th' cs'"
   shows "waiting (e#s) th' cs'"
@@ -1749,7 +1485,7 @@
   by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2) 
       rotate1.simps(2) self_append_conv2 set_rotate1 
         th_not_in_wq waiting_eq wq_es_cs wq_neq_simp)
-  
+
 lemma holding_kept:
   assumes "holding s th' cs'"
   shows "holding (e#s) th' cs'"
@@ -1767,116 +1503,16 @@
   thus ?thesis
     by (simp add: cs_holding_def holding_eq) 
 qed
-
-end
-
-
-context valid_trace_p_w
-begin
-
-lemma wq_s_cs: "wq s cs = holder#waiters"
-    by (simp add: holder_def waiters_def wne)
-    
-lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
-  by (simp add: wq_es_cs wq_s_cs)
-
-lemma waiting_es_th_cs: "waiting (e#s) th cs"
-  using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
-
-lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
-   by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
-
-lemma holding_esE:
-  assumes "holding (e#s) th' cs'"
-  obtains "holding s th' cs'"
-  using assms 
-proof(cases "cs' = cs")
-  case False
-  hence "wq (e#s) cs' = wq s cs'" by simp
-  with assms show ?thesis
-    using cs_holding_def holding_eq that by auto 
-next
-  case True
-  with assms show ?thesis
-  by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that 
-        wq_es_cs' wq_s_cs) 
-qed
-
-lemma waiting_esE:
-  assumes "waiting (e#s) th' cs'"
-  obtains "th' \<noteq> th" "waiting s th' cs'"
-     |  "th' = th" "cs' = cs"
-proof(cases "waiting s th' cs'")
-  case True
-  have "th' \<noteq> th"
-  proof
-    assume otherwise: "th' = th"
-    from True[unfolded this]
-    show False by (simp add: th_not_waiting) 
-  qed
-  from that(1)[OF this True] show ?thesis .
-next
-  case False
-  hence "th' = th \<and> cs' = cs"
-      by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) 
-        set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
-  with that(2) show ?thesis by metis
+end 
+
+lemma (in valid_trace_p) 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 RAG_es: "RAG (e # s) =  RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
-proof(rule rel_eqI)
-  fix n1 n2
-  assume "(n1, n2) \<in> ?L"
-  thus "(n1, n2) \<in> ?R" 
-  proof(cases rule:in_RAG_E)
-    case (waiting th' cs')
-    from this(3)
-    show ?thesis
-    proof(cases rule:waiting_esE)
-      case 1
-      thus ?thesis using waiting(1,2)
-        by (unfold s_RAG_def, fold waiting_eq, auto)
-    next
-      case 2
-      thus ?thesis using waiting(1,2) by auto
-    qed
-  next
-    case (holding th' cs')
-    from this(3)
-    show ?thesis
-    proof(cases rule:holding_esE)
-      case 1
-      with holding(1,2)
-      show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-  qed
-next
-  fix n1 n2
-  assume "(n1, n2) \<in> ?R"
-  hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
-  thus "(n1, n2) \<in> ?L"
-  proof
-    assume "(n1, n2) \<in> RAG s"
-    thus ?thesis
-    proof(cases rule:in_RAG_E)
-      case (waiting th' cs')
-      from waiting_kept[OF this(3)]
-      show ?thesis using waiting(1,2)
-         by (unfold s_RAG_def, fold waiting_eq, auto)
-    next
-      case (holding th' cs')
-      from holding_kept[OF this(3)]
-      show ?thesis using holding(1,2)
-         by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-  next
-    assume "n1 = Th th \<and> n2 = Cs cs"
-    thus ?thesis using RAG_edge by auto
-  qed
-qed
-
-end
-
 context valid_trace_p_h
 begin
 
@@ -1973,10 +1609,116 @@
 
 end
 
+context valid_trace_p_w
+begin
+
+lemma wq_s_cs: "wq s cs = holder#waiters"
+    by (simp add: holder_def waiters_def wne)
+    
+lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
+  by (simp add: wq_es_cs wq_s_cs)
+
+lemma waiting_es_th_cs: "waiting (e#s) th cs"
+  using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
+
+lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
+   by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
+
+lemma holding_esE:
+  assumes "holding (e#s) th' cs'"
+  obtains "holding s th' cs'"
+  using assms 
+proof(cases "cs' = cs")
+  case False
+  hence "wq (e#s) cs' = wq s cs'" by simp
+  with assms show ?thesis
+    using cs_holding_def holding_eq that by auto 
+next
+  case True
+  with assms show ?thesis
+  by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that 
+        wq_es_cs' wq_s_cs) 
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) th' cs'"
+  obtains "th' \<noteq> th" "waiting s th' cs'"
+     |  "th' = th" "cs' = cs"
+proof(cases "waiting s th' cs'")
+  case True
+  have "th' \<noteq> th"
+  proof
+    assume otherwise: "th' = th"
+    from True[unfolded this]
+    show False by (simp add: th_not_waiting)
+  qed
+  from that(1)[OF this True] show ?thesis .
+next
+  case False
+  hence "th' = th \<and> cs' = cs"
+      by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) 
+        set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
+  with that(2) show ?thesis by metis
+qed
+
+lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R" 
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:waiting_esE)
+      case 1
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case 2
+      thus ?thesis using waiting(1,2) by auto
+    qed
+  next
+    case (holding th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:holding_esE)
+      case 1
+      with holding(1,2)
+      show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  qed
+next
+  fix n1 n2
+  assume "(n1, n2) \<in> ?R"
+  hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
+  thus "(n1, n2) \<in> ?L"
+  proof
+    assume "(n1, n2) \<in> RAG s"
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from waiting_kept[OF this(3)]
+      show ?thesis using waiting(1,2)
+         by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from holding_kept[OF this(3)]
+      show ?thesis using holding(1,2)
+         by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  next
+    assume "n1 = Th th \<and> n2 = Cs cs"
+    thus ?thesis using RAG_edge by auto
+  qed
+qed
+
+end
+
 context valid_trace_p
 begin
 
-lemma RAG_es': "RAG (e # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+lemma RAG_es: "RAG (e # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
                                                   else RAG s \<union> {(Th th, Cs cs)})"
 proof(cases "wq s cs = []")
   case True
@@ -1992,6 +1734,252 @@
 
 end
 
+section {* Finiteness of RAG *}
+
+context valid_trace
+begin
+
+lemma finite_RAG:
+  shows "finite (RAG s)"
+proof(induct rule:ind)
+  case Nil
+  show ?case 
+  by (auto simp: s_RAG_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+next
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case
+  proof(cases e)
+    case (Create th prio)
+    interpret vt: valid_trace_create s e th prio using Create
+      by (unfold_locales, simp)
+    show ?thesis using Cons by simp
+  next
+    case (Exit th)
+    interpret vt: valid_trace_exit s e th using Exit
+      by (unfold_locales, simp)
+    show ?thesis using Cons by simp
+  next
+    case (P th cs)
+    interpret vt: valid_trace_p s e th cs using P
+      by (unfold_locales, simp)
+    show ?thesis using Cons using vt.RAG_es by auto 
+  next
+    case (V th cs)
+    interpret vt: valid_trace_v s e th cs using V
+      by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt.finite_RAG_kept) 
+  next
+    case (Set th prio)
+    interpret vt: valid_trace_set s e th prio using Set
+      by (unfold_locales, simp)
+    show ?thesis using Cons by simp
+  qed
+qed
+end
+
+section {* RAG is acyclic *}
+
+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.
+*}
+
+
+context valid_trace
+begin
+
+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" 
+    interpret ve2: valid_moment_e _ t2 using lt2
+     by (unfold_locales, simp)
+    let ?e = ve2.next_e
+    have "t2 < ?t3" by simp
+    from nn2 [rule_format, OF this] and ve2.trace_e
+    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 ?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 
+          thus ?thesis
+            using True h2 ve2.vat_moment_e.wq_out_inv by blast 
+        qed
+        thus ?thesis
+          using step.cases ve2.vat_moment_e.pip_e by auto 
+      next
+        case False
+        hence "?e = P thread cs2"
+          using h1 ve2.vat_moment_e.wq_in_inv by blast 
+        thus ?thesis
+          using step.cases ve2.vat_moment_e.pip_e 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"
+      interpret ve2: valid_moment_e _ t2 using lt2
+        by (unfold_locales, simp)
+      let ?e = ve2.next_e
+      have "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and ve2.trace_e
+      have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" by auto
+      have lt_2: "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and ve2.trace_e
+      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], unfolded ve2.trace_e[folded eq_12]] 
+           eq_12[symmetric]
+      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 "?e = V thread cs2 \<or> ?e = P thread cs2"
+        using h1 h2 np2 ve2.vat_moment_e.wq_in_inv 
+              ve2.vat_moment_e.wq_out_inv by blast
+      moreover have "?e = V thread cs1 \<or> ?e = P thread cs1"
+        using eq_12 g1 g2 np1 ve2.vat_moment_e.wq_in_inv 
+              ve2.vat_moment_e.wq_out_inv by blast
+      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
+
+lemma (in valid_trace_v)
+  preced_es [simp]: "preced th (e#s) = preced th s"
+  by (unfold is_v preced_def, simp)
+
+lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
+proof
+  fix th'
+  show "the_preced (V th cs # s) th' = the_preced s th'"
+    by (unfold the_preced_def preced_def, simp)
+qed
+
+
+lemma (in valid_trace_v)
+  the_preced_es: "the_preced (e#s) = the_preced s"
+  by (unfold is_v preced_def, simp)
+
+context valid_trace_p
+begin
+
+lemma not_holding_s_th_cs: "\<not> holding s th cs"
+proof
+  assume otherwise: "holding s th cs"
+  from pip_e[unfolded is_p]
+  show False
+  proof(cases)
+    case (thread_P)
+    moreover have "(Cs cs, Th th) \<in> RAG s"
+      using otherwise cs_holding_def 
+            holding_eq th_not_in_wq by auto
+    ultimately show ?thesis by auto
+  qed
+qed
+
+end
+
+
 lemma (in valid_trace_v_n) finite_waiting_set:
   "finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
     by (simp add: waiting_set_eq)
@@ -2008,28 +1996,6 @@
   "finite {(Cs cs, Th th') |th'. next_th s th cs th'}"
     by (simp add: holding_set_eq)
 
-context valid_trace_v
-begin
-
-lemma 
-  finite_RAG_kept:
-  assumes "finite (RAG s)"
-  shows "finite (RAG (e#s))"
-proof(cases "rest = []")
-  case True
-  interpret vt: valid_trace_v_e using True
-    by (unfold_locales, simp)
-  show ?thesis using assms
-    by  (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
-next
-  case False
-  interpret vt: valid_trace_v_n using False
-    by (unfold_locales, simp)
-  show ?thesis using assms
-    by  (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
-qed
-
-end
 
 context valid_trace_v_e
 begin 
@@ -2074,7 +2040,7 @@
         by (unfold s_RAG_def, auto)
       from this(2) have "waiting s taker cs'" 
         by (unfold s_RAG_def, fold waiting_eq, auto)
-      from waiting_unique[OF this waiting_taker]
+      from waiting_unique[OF this waiting_taker] 
       have "cs' = cs" .
       from h(1)[unfolded this] show False by auto
     qed
@@ -2109,7 +2075,7 @@
         by (unfold s_RAG_def, auto)
       hence "waiting s th cs'" 
         by (unfold s_RAG_def, fold waiting_eq, auto)
-      with th_not_waiting show False by auto
+      with th_not_waiting show False by auto (* ccc *)
     qed
     ultimately show ?thesis by auto
   qed
@@ -2153,45 +2119,6 @@
 context valid_trace
 begin
 
-lemma finite_RAG:
-  shows "finite (RAG s)"
-proof(induct rule:ind)
-  case Nil
-  show ?case 
-  by (auto simp: s_RAG_def cs_waiting_def 
-                   cs_holding_def wq_def acyclic_def)
-next
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case
-  proof(cases e)
-    case (Create th prio)
-    interpret vt: valid_trace_create s e th prio using Create
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
-  next
-    case (Exit th)
-    interpret vt: valid_trace_exit s e th using Exit
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged)
-  next
-    case (P th cs)
-    interpret vt: valid_trace_p s e th cs using P
-      by (unfold_locales, simp)
-    show ?thesis using Cons using vt.RAG_es' by auto 
-  next
-    case (V th cs)
-    interpret vt: valid_trace_v s e th cs using V
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.finite_RAG_kept) 
-  next
-    case (Set th prio)
-    interpret vt: valid_trace_set s e th prio using Set
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
-  qed
-qed
-
 lemma acyclic_RAG:
   shows "acyclic (RAG s)"
 proof(induct rule:ind)
@@ -2207,12 +2134,12 @@
     case (Create th prio)
     interpret vt: valid_trace_create s e th prio using Create
       by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
+    show ?thesis using Cons by simp 
   next
     case (Exit th)
     interpret vt: valid_trace_exit s e th using Exit
       by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged)
+    show ?thesis using Cons by simp
   next
     case (P th cs)
     interpret vt: valid_trace_p s e th cs using P
@@ -2249,10 +2176,31 @@
     case (Set th prio)
     interpret vt: valid_trace_set s e th prio using Set
       by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
+    show ?thesis using Cons by simp 
   qed
 qed
 
+end
+
+section {* RAG is single-valued *}
+
+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 held_unique)
+
+lemma sgv_RAG: "single_valued (RAG s)"
+  using unique_RAG by (auto simp:single_valued_def)
+
+end
+
+section {* RAG is well-founded *}
+
+context valid_trace
+begin
+
 lemma wf_RAG: "wf (RAG s)"
 proof(rule finite_acyclic_wf)
   from finite_RAG show "finite (RAG s)" .
@@ -2260,6 +2208,59 @@
   from acyclic_RAG show "acyclic (RAG s)" .
 qed
 
+lemma wf_RAG_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
+
+section {* RAG forms a forest (or tree) *}
+
+context valid_trace
+begin
+
+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"
+  using rtree_RAG .
+
+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
+
+section {* Derived properties for parts of RAG *}
+
+context valid_trace
+begin
+
+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_wRAG: "single_valued (wRAG s)"
   using waiting_unique
   by (unfold single_valued_def wRAG_def, auto)
@@ -2272,39 +2273,8 @@
   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 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 held_unique)
-
-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)" .
@@ -2312,24 +2282,6 @@
   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
-
-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)
-
 sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
 proof -
   have "fsubtree (tRAG s)"
@@ -2363,6 +2315,70 @@
 qed
 
 
+(* ccc *)
+
+context valid_trace_p 
+begin
+
+lemma ready_th_s: "th \<in> readys s"
+  using runing_th_s
+  by (unfold runing_def, auto)
+
+lemma live_th_s: "th \<in> threads s"
+  using readys_threads ready_th_s by auto
+
+lemma live_th_es: "th \<in> threads (e#s)"
+  using live_th_s 
+  by (unfold is_p, simp)
+
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+end
+
+context valid_trace_v
+begin
+
+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 
+
+end
+
+
+context valid_trace_e 
+begin
+
+lemma actor_inv: 
+  assumes "\<not> isCreate e"
+  shows "actor e \<in> runing s"
+  using pip_e assms 
+  by (induct, auto)
+
+end
+
+
+(* ccc *)
+
+(* drag more from before to here *)
+
+
+section {* ccc *}
+
+
 context valid_trace
 begin
 
@@ -2393,10 +2409,9 @@
       by (simp add: subtree_def the_preced_def)   
   qed
 
-
-lemma (in valid_trace) finite_threads:
+lemma  finite_threads:
   shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
+  using vt by (induct) (auto elim: step.cases)
 
 lemma cp_le:
   assumes th_in: "th \<in> threads s"
@@ -2432,16 +2447,6 @@
   ultimately show ?thesis by auto
 qed
 
-lemma wf_RAG_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
-
 lemma chain_building:
   assumes "node \<in> Domain (RAG s)"
   obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"
@@ -2487,7 +2492,7 @@
 
 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)^+)"
@@ -2585,13 +2590,6 @@
   apply (unfold children_RAG_alt_def cntCS_def holdents_def)
   by (rule card_image[symmetric], auto simp:inj_on_def)
 
-context valid_trace
-begin
-
-lemma finite_holdents: "finite (holdents s th)"
-  by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)
-  
-end
 
 context valid_trace_p_w
 begin
@@ -2655,6 +2653,9 @@
 
 end
   
+lemma (in valid_trace) finite_holdents: "finite (holdents s th)"
+  by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)
+
 context valid_trace_p_h
 begin
 
@@ -2874,7 +2875,7 @@
 end
 
 
-context valid_trace_v (* ccc *)
+context valid_trace_v 
 begin
 
 lemma holding_th_cs_s: 
@@ -2905,7 +2906,7 @@
 proof -
   have "cs \<in> holdents s th" using holding_th_cs_s
     by (unfold holdents_def, simp)
-  moreover have "finite (holdents s th)" using finite_holdents
+  moreover have "finite (holdents s th)" using finite_holdents (* ccc *)
     by simp
   ultimately show ?thesis
     by (unfold cntCS_def, 
@@ -4630,10 +4631,6 @@
 context valid_trace
 begin
 
-thm th_chain_to_ready
-
-find_theorems subtree Th RAG
-
 lemma threads_alt_def:
   "(threads s) = (\<Union> th \<in> readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)})"
     (is "?L = ?R")
diff -r f7b33c633b96 -r 3d2b59f15f26 PIPDefs.thy
--- a/PIPDefs.thy	Sun Jan 31 18:15:13 2016 +0800
+++ b/PIPDefs.thy	Mon Feb 01 20:56:39 2016 +0800
@@ -657,6 +657,11 @@
   by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv 
              s_holding_abv cs_RAG_def, auto)
 
+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)
+
 definition "cp_gen s x =
                   Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"