# HG changeset patch
# User zhangx
# Date 1454035912 -28800
# Node ID 4763aa246dbd7bee323ee0e394bc8cb77291e033
# Parent  0525670d8e6aaf211669e9bae7d525e825de3546
Original files overwrite by their parallels (Correctness.thy v.s. PrioG.thy, PIPBasics.thy v.s. CpsG.thy, Implementation v.s. ExtGG.thy).

diff -r 0525670d8e6a -r 4763aa246dbd Correctness.thy
--- a/Correctness.thy	Fri Jan 29 09:46:07 2016 +0800
+++ b/Correctness.thy	Fri Jan 29 10:51:52 2016 +0800
@@ -1,8 +1,7 @@
-theory Correctness
-imports PIPBasics
+theory PrioG
+imports CpsG
 begin
 
-
 text {* 
   The following two auxiliary lemmas are used to reason about @{term Max}.
 *}
@@ -474,40 +473,45 @@
 section {* The `blocking thread` *}
 
 text {* 
+  The purpose of PIP is to ensure that the most 
+  urgent thread @{term th} is not blocked unreasonably. 
+  Therefore, a clear picture of the blocking thread is essential 
+  to assure people that the purpose is fulfilled. 
+  
+  In this section, we are going to derive a series of lemmas 
+  with finally give rise to a picture of the blocking thread. 
 
-  The purpose of PIP is to ensure that the most urgent thread @{term
-  th} is not blocked unreasonably. Therefore, below, we will derive
-  properties of the blocking thread. By blocking thread, we mean a
-  thread in running state t @ s, but is different from thread @{term
-  th}.
-
-  The first lemmas shows that the @{term cp}-value of the blocking
-  thread @{text th'} equals to the highest precedence in the whole
-  system.
-
+  By `blocking thread`, we mean a thread in running state but 
+  different from thread @{term th}.
 *}
 
+text {*
+  The following lemmas shows that the @{term cp}-value 
+  of the blocking thread @{text th'} equals to the highest
+  precedence in the whole system.
+*}
 lemma runing_preced_inversion:
-  assumes runing': "th' \<in> runing (t @ s)"
-  shows "cp (t @ s) th' = preced th s" 
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "cp (t@s) th' = preced th s" (is "?L = ?R")
 proof -
-  have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))" 
-    using assms by (unfold runing_def, auto)
-  also have "\<dots> = preced th s"
-    by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) 
+  have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
+      by (unfold runing_def, auto)
+  also have "\<dots> = ?R"
+      by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) 
   finally show ?thesis .
 qed
 
 text {*
 
-  The next lemma shows how the counters for @{term "P"} and @{term
-  "V"} operations relate to the running threads in the states @{term
-  s} and @{term "t @ s"}: if a thread's @{term "P"}-count equals its
-  @{term "V"}-count (which means it no longer has any resource in its
-  possession), it cannot be a running thread.
+  The following lemma shows how the counters for @{term "P"} and
+  @{term "V"} operations relate to the running threads in the states
+  @{term s} and @{term "t @ s"}.  The lemma shows that if a thread's
+  @{term "P"}-count equals its @{term "V"}-count (which means it no
+  longer has any resource in its possession), it cannot be a running
+  thread.
 
   The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
-  The key is the use of @{thm count_eq_dependants} to derive the
+  The key is the use of @{thm eq_pv_dependants} to derive the
   emptiness of @{text th'}s @{term dependants}-set from the balance of
   its @{term P} and @{term V} counts.  From this, it can be shown
   @{text th'}s @{term cp}-value equals to its own precedence.
@@ -516,7 +520,7 @@
   runing_preced_inversion}, its @{term cp}-value equals to the
   precedence of @{term th}.
 
-  Combining the above two results we have that @{text th'} and @{term
+  Combining the above two resukts we have that @{text th'} and @{term
   th} have the same precedence. By uniqueness of precedences, we have
   @{text "th' = th"}, which is in contradiction with the assumption
   @{text "th' \<noteq> th"}.
@@ -525,13 +529,13 @@
                       
 lemma eq_pv_blocked: (* ddd *)
   assumes neq_th': "th' \<noteq> th"
-  and eq_pv: "cntP (t @ s) th' = cntV (t @ s) th'"
-  shows "th' \<notin> runing (t @ s)"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
 proof
-  assume otherwise: "th' \<in> runing (t @ s)"
+  assume otherwise: "th' \<in> runing (t@s)"
   show False
   proof -
-    have th'_in: "th' \<in> threads (t @ s)"
+    have th'_in: "th' \<in> threads (t@s)"
         using otherwise readys_threads runing_def by auto 
     have "th' = th"
     proof(rule preced_unique)
@@ -545,12 +549,13 @@
         -- {* Since the counts of @{term th'} are balanced, the subtree
               of it contains only itself, so, its @{term cp}-value
               equals its @{term preced}-value: *}
-        have "?L = cp (t @ s) th'"
-          by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
+        have "?L = cp (t@s) th'"
+         by (unfold cp_eq_cpreced cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp)
         -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion},
               its @{term cp}-value equals @{term "preced th s"}, 
               which equals to @{term "?R"} by simplification: *}
         also have "... = ?R" 
+        thm runing_preced_inversion
             using runing_preced_inversion[OF otherwise] by simp
         finally show ?thesis .
       qed
@@ -568,8 +573,8 @@
 lemma eq_pv_persist: (* ddd *)
   assumes neq_th': "th' \<noteq> th"
   and eq_pv: "cntP s th' = cntV s th'"
-  shows "cntP (t @ s) th' = cntV (t @ s) th'"
-proof(induction rule: ind) 
+  shows "cntP (t@s) th' = cntV (t@s) th'"
+proof(induction rule:ind) -- {* The proof goes by induction. *}
   -- {* The nontrivial case is for the @{term Cons}: *}
   case (Cons e t)
   -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *}
@@ -618,28 +623,22 @@
 qed (auto simp:eq_pv)
 
 text {*
-
-  By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist}, it can
-  be derived easily that @{term th'} can not be running in the future:
-
+  By combining @{thm  eq_pv_blocked} and @{thm eq_pv_persist},
+  it can be derived easily that @{term th'} can not be running in the future:
 *}
-
 lemma eq_pv_blocked_persist:
   assumes neq_th': "th' \<noteq> th"
   and eq_pv: "cntP s th' = cntV s th'"
-  shows "th' \<notin> runing (t @ s)"
+  shows "th' \<notin> runing (t@s)"
   using assms
   by (simp add: eq_pv_blocked eq_pv_persist) 
 
 text {*
-
-  The following lemma shows the blocking thread @{term th'} must hold
-  some resource in the very beginning.
-
+  The following lemma shows the blocking thread @{term th'}
+  must hold some resource in the very beginning. 
 *}
-
 lemma runing_cntP_cntV_inv: (* ddd *)
-  assumes is_runing: "th' \<in> runing (t @ s)"
+  assumes is_runing: "th' \<in> runing (t@s)"
   and neq_th': "th' \<noteq> th"
   shows "cntP s th' > cntV s th'"
   using assms
@@ -665,13 +664,11 @@
 
 
 text {*
+  The following lemmas shows the blocking thread @{text th'} must be live 
+  at the very beginning, i.e. the moment (or state) @{term s}. 
 
-  The following lemmas shows the blocking thread @{text th'} must be
-  live at the very beginning, i.e. the moment (or state) @{term s}.
   The proof is a  simple combination of the results above:
-
 *}
-
 lemma runing_threads_inv: 
   assumes runing': "th' \<in> runing (t@s)"
   and neq_th': "th' \<noteq> th"
@@ -689,12 +686,9 @@
 qed
 
 text {*
-
-  The following lemma summarises the above lemmas to give an overall
-  characterisationof the blocking thread @{text "th'"}:
-
+  The following lemma summarizes several foregoing 
+  lemmas to give an overall picture of the blocking thread @{text "th'"}:
 *}
-                  
 lemma runing_inversion: (* ddd, one of the main lemmas to present *)
   assumes runing': "th' \<in> runing (t@s)"
   and neq_th: "th' \<noteq> th"
@@ -712,23 +706,18 @@
   show "cp (t@s) th' = preced th s" .
 qed
 
-
 section {* The existence of `blocking thread` *}
 
 text {* 
-
-  Suppose @{term th} is not running, it is first shown that there is a
-  path in RAG leading from node @{term th} to another thread @{text
-  "th'"} in the @{term readys}-set (So @{text "th'"} is an ancestor of
-  @{term th}}).
+  Suppose @{term th} is not running, it is first shown that
+  there is a path in RAG leading from node @{term th} to another thread @{text "th'"} 
+  in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
 
-  Now, since @{term readys}-set is non-empty, there must be one in it
-  which holds the highest @{term cp}-value, which, by definition, is
-  the @{term runing}-thread. However, we are going to show more: this
-  running thread is exactly @{term "th'"}.
-
-*}
-
+  Now, since @{term readys}-set is non-empty, there must be
+  one in it which holds the highest @{term cp}-value, which, by definition, 
+  is the @{term runing}-thread. However, we are going to show more: this running thread
+  is exactly @{term "th'"}.
+     *}
 lemma th_blockedE: (* ddd, the other main lemma to be presented: *)
   assumes "th \<notin> runing (t@s)"
   obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
@@ -760,7 +749,7 @@
         show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
       next
         show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
-          by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) 
+          by (metis Range.intros dp trancl_range vat_t.rg_RAG_threads vat_t.subtree_tRAG_thread) 
       next
         show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
                     by (unfold tRAG_subtree_eq, auto simp:subtree_def)
@@ -790,15 +779,11 @@
 qed
 
 text {*
-
-  Now it is easy to see there is always a thread to run by case
-  analysis on whether thread @{term th} is running: if the answer is
-  yes, the the running thread is obviously @{term th} itself;
-  otherwise, the running thread is the @{text th'} given by lemma
-  @{thm th_blockedE}.
-
+  Now it is easy to see there is always a thread to run by case analysis
+  on whether thread @{term th} is running: if the answer is Yes, the 
+  the running thread is obviously @{term th} itself; otherwise, the running
+  thread is the @{text th'} given by lemma @{thm th_blockedE}.
 *}
-
 lemma live: "runing (t@s) \<noteq> {}"
 proof(cases "th \<in> runing (t@s)") 
   case True thus ?thesis by auto
@@ -807,6 +792,5 @@
   thus ?thesis using th_blockedE by auto
 qed
 
-
 end
 end
diff -r 0525670d8e6a -r 4763aa246dbd Implementation.thy
--- a/Implementation.thy	Fri Jan 29 09:46:07 2016 +0800
+++ b/Implementation.thy	Fri Jan 29 10:51:52 2016 +0800
@@ -2,8 +2,8 @@
   This file contains lemmas used to guide the recalculation of current precedence 
   after every system call (or system operation)
 *}
-theory Implementation
-imports PIPBasics
+theory ExtGG
+imports CpsG
 begin
 
 text {* (* ddd *)
@@ -35,30 +35,7 @@
 
 section {* The @{term Set} operation *}
 
-text {* (* ddd *)
-  The following locale @{text "step_set_cps"} investigates the recalculation 
-  after the @{text "Set"} operation.
-*}
-locale step_set_cps =
-  fixes s' th prio s 
-  -- {* @{text "s'"} is the system state before the operation *}
-  -- {* @{text "s"} is the system state after the operation *}
-  defines s_def : "s \<equiv> (Set th prio#s')" 
-  -- {* @{text "s"} is assumed to be a legitimate state, from which
-         the legitimacy of @{text "s"} can be derived. *}
-  assumes vt_s: "vt s"
-
-sublocale step_set_cps < vat_s : valid_trace "s"
-proof
-  from vt_s show "vt s" .
-qed
-
-sublocale step_set_cps < vat_s' : valid_trace "s'"
-proof
-  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_set_cps 
+context valid_trace_set
 begin
 
 text {* (* ddd *)
@@ -67,26 +44,21 @@
   of the operation (or event).
 *}
 
+
 lemma eq_preced:
   assumes "th' \<noteq> th"
-  shows "preced th' s = preced th' s'"
+  shows "preced th' (e#s) = preced th' s"
 proof -
   from assms show ?thesis 
-    by (unfold s_def, auto simp:preced_def)
+    by (unfold is_set, auto simp:preced_def)
 qed
 
 lemma eq_the_preced: 
   assumes "th' \<noteq> th"
-  shows "the_preced s th' = the_preced s' th'"
+  shows "the_preced (e#s) th' = the_preced s th'"
   using assms
   by (unfold the_preced_def, intro eq_preced, simp)
 
-text {*
-  The following lemma assures that the resetting of priority does not change the RAG. 
-*}
-
-lemma eq_dep: "RAG s = RAG s'"
-  by (unfold s_def RAG_set_unchanged, auto)
 
 text {* (* ddd *)
   Th following lemma @{text "eq_cp_pre"} says that the priority change of @{text "th"}
@@ -97,18 +69,18 @@
 *}
 
 lemma eq_cp_pre:
-  assumes nd: "Th th \<notin> subtree (RAG s') (Th th')"
-  shows "cp s th' = cp s' th'"
+  assumes nd: "Th th \<notin> subtree (RAG s) (Th th')"
+  shows "cp (e#s) th' = cp s th'"
 proof -
   -- {* After unfolding using the alternative definition, elements 
         affecting the @{term "cp"}-value of threads become explicit. 
         We only need to prove the following: *}
-  have "Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
-        Max (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+  have "Max (the_preced (e#s) ` {th'a. Th th'a \<in> subtree (RAG (e#s)) (Th th')}) =
+        Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')})"
         (is "Max (?f ` ?S1) = Max (?g ` ?S2)")
   proof -
     -- {* The base sets are equal. *}
-    have "?S1 = ?S2" using eq_dep by simp
+    have "?S1 = ?S2" using RAG_unchanged by simp
     -- {* The function values on the base set are equal as well. *}
     moreover have "\<forall> e \<in> ?S2. ?f e = ?g e"
     proof
@@ -116,7 +88,7 @@
       assume "th1 \<in> ?S2"
       with nd have "th1 \<noteq> th" by (auto)
       from eq_the_preced[OF this]
-      show "the_preced s th1 = the_preced s' th1" .
+      show "the_preced (e#s) th1 = the_preced s th1" .
     qed
     -- {* Therefore, the image of the functions are equal. *}
     ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq)
@@ -131,16 +103,9 @@
 *}
 lemma th_in_no_subtree:
   assumes "th' \<noteq> th"
-  shows "Th th \<notin> subtree (RAG s') (Th th')"
+  shows "Th th \<notin> subtree (RAG s) (Th th')"
 proof -
-  have "th \<in> readys s'"
-  proof -
-    from step_back_step [OF vt_s[unfolded s_def]]
-    have "step s' (Set th prio)" .
-    hence "th \<in> runing s'" by (cases, simp)
-    thus ?thesis by (simp add:readys_def runing_def)
-  qed
-  from vat_s'.readys_in_no_subtree[OF this assms(1)]
+  from readys_in_no_subtree[OF th_ready_s assms(1)]
   show ?thesis by blast
 qed
 
@@ -151,7 +116,7 @@
 *}
 lemma eq_cp:
   assumes "th' \<noteq> th"
-  shows "cp s th' = cp s' th'"
+  shows "cp (e#s) th' = cp s th'"
   by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])
 
 end
@@ -162,73 +127,33 @@
   The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.
 *}
 
-locale step_v_cps =
-  -- {* @{text "th"} is the initiating thread *}
-  -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *}
-  fixes s' th cs s    -- {* @{text "s'"} is the state before operation*}
-  defines s_def : "s \<equiv> (V th cs#s')" -- {* @{text "s"} is the state after operation*}
-  -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *}
-  assumes vt_s: "vt s"
 
-sublocale step_v_cps < vat_s : valid_trace "s"
-proof
-  from vt_s show "vt s" .
-qed
-
-sublocale step_v_cps < vat_s' : valid_trace "s'"
-proof
-  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_v_cps
+context valid_trace_v
 begin
 
-lemma ready_th_s': "th \<in> readys s'"
-  using step_back_step[OF vt_s[unfolded s_def]]
-  by (cases, simp add:runing_def)
-
-lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"
+lemma ancestors_th: "ancestors (RAG s) (Th th) = {}"
 proof -
-  from vat_s'.readys_root[OF ready_th_s']
+  from readys_root[OF th_ready_s]
   show ?thesis
   by (unfold root_def, simp)
 qed
 
-lemma holding_th: "holding s' th cs"
+lemma edge_of_th:
+    "(Cs cs, Th th) \<in> RAG s" 
 proof -
-  from vt_s[unfolded s_def]
-  have " PIP s' (V th cs)" by (cases, simp)
-  thus ?thesis by (cases, auto)
-qed
-
-lemma edge_of_th:
-    "(Cs cs, Th th) \<in> RAG s'" 
-proof -
- from holding_th
+ from holding_th_cs_s
  show ?thesis 
     by (unfold s_RAG_def holding_eq, auto)
 qed
 
 lemma ancestors_cs: 
-  "ancestors (RAG s') (Cs cs) = {Th th}"
+  "ancestors (RAG s) (Cs cs) = {Th th}"
 proof -
-  have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th)  \<union>  {Th th}"
-  proof(rule vat_s'.rtree_RAG.ancestors_accum)
-    from vt_s[unfolded s_def]
-    have " PIP s' (V th cs)" by (cases, simp)
-    thus "(Cs cs, Th th) \<in> RAG s'" 
-    proof(cases)
-      assume "holding s' th cs"
-      from this[unfolded holding_eq]
-      show ?thesis by (unfold s_RAG_def, auto)
-    qed
-  qed
+  have "ancestors (RAG s) (Cs cs) = ancestors (RAG s) (Th th)  \<union>  {Th th}"
+   by (rule rtree_RAG.ancestors_accum[OF edge_of_th])
   from this[unfolded ancestors_th] show ?thesis by simp
 qed
 
-lemma preced_kept: "the_preced s = the_preced s'"
-  by (auto simp: s_def the_preced_def preced_def)
-
 end
 
 text {*
@@ -236,147 +161,99 @@
   which represents the case when there is another thread @{text "th'"}
   to take over the critical resource released by the initiating thread @{text "th"}.
 *}
-locale step_v_cps_nt = step_v_cps +
-  fixes th'
-  -- {* @{text "th'"} is assumed to take over @{text "cs"} *}
-  assumes nt: "next_th s' th cs th'" 
 
-context step_v_cps_nt
+context valid_trace_v_n
 begin
 
-text {*
-  Lemma @{text "RAG_s"} confirms the change of RAG:
-  two edges removed and one added, as shown by the following diagram.
-*}
-
-(*
-  RAG before the V-operation
-    th1 ----|
-            |
-    th' ----|
-            |----> cs -----|
-    th2 ----|              |
-            |              |
-    th3 ----|              |
-                           |------> th
-    th4 ----|              |
-            |              |
-    th5 ----|              |
-            |----> cs'-----|
-    th6 ----|
-            |
-    th7 ----|
-
- RAG after the V-operation
-    th1 ----|
-            |
-            |----> cs ----> th'
-    th2 ----|              
-            |              
-    th3 ----|              
-                           
-    th4 ----|              
-            |              
-    th5 ----|              
-            |----> cs'----> th
-    th6 ----|
-            |
-    th7 ----|
-*)
-
-lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"
-                using next_th_RAG[OF nt]  .
+lemma sub_RAGs': 
+  "{(Cs cs, Th th), (Th taker, Cs cs)} \<subseteq> RAG s"
+     using next_th_RAG[OF next_th_taker]  .
 
 lemma ancestors_th': 
-  "ancestors (RAG s') (Th th') = {Th th, Cs cs}" 
+  "ancestors (RAG s) (Th taker) = {Th th, Cs cs}" 
 proof -
-  have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"
-  proof(rule  vat_s'.rtree_RAG.ancestors_accum)
-    from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto
+  have "ancestors (RAG s) (Th taker) = ancestors (RAG s) (Cs cs) \<union> {Cs cs}"
+  proof(rule  rtree_RAG.ancestors_accum)
+    from sub_RAGs' show "(Th taker, Cs cs) \<in> RAG s" by auto
   qed
   thus ?thesis using ancestors_th ancestors_cs by auto
 qed
 
 lemma RAG_s:
-  "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
-                                         {(Cs cs, Th th')}"
-proof -
-  from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
-    and nt show ?thesis  by (auto intro:next_th_unique)
-qed
+  "RAG (e#s) = (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) \<union>
+                                         {(Cs cs, Th taker)}"
+ by (unfold RAG_es waiting_set_eq holding_set_eq, auto)
 
 lemma subtree_kept: (* ddd *)
-  assumes "th1 \<notin> {th, th'}"
-  shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R")
+  assumes "th1 \<notin> {th, taker}"
+  shows "subtree (RAG (e#s)) (Th th1) = 
+                     subtree (RAG s) (Th th1)" (is "_ = ?R")
 proof -
-  let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})"
-  let ?RAG'' = "?RAG' \<union> {(Cs cs, Th th')}"
+  let ?RAG' = "(RAG s - {(Cs cs, Th th), (Th taker, Cs cs)})"
+  let ?RAG'' = "?RAG' \<union> {(Cs cs, Th taker)}"
   have "subtree ?RAG' (Th th1) = ?R" 
   proof(rule subset_del_subtree_outside)
-    show "Range {(Cs cs, Th th), (Th th', Cs cs)} \<inter> subtree (RAG s') (Th th1) = {}"
+    show "Range {(Cs cs, Th th), (Th taker, Cs cs)} \<inter> subtree (RAG s) (Th th1) = {}"
     proof -
-      have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+      have "(Th th) \<notin> subtree (RAG s) (Th th1)"
       proof(rule subtree_refute)
-        show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+        show "Th th1 \<notin> ancestors (RAG s) (Th th)"
           by (unfold ancestors_th, simp)
       next
         from assms show "Th th1 \<noteq> Th th" by simp
       qed
-      moreover have "(Cs cs) \<notin>  subtree (RAG s') (Th th1)"
+      moreover have "(Cs cs) \<notin>  subtree (RAG s) (Th th1)"
       proof(rule subtree_refute)
-        show "Th th1 \<notin> ancestors (RAG s') (Cs cs)"
+        show "Th th1 \<notin> ancestors (RAG s) (Cs cs)"
           by (unfold ancestors_cs, insert assms, auto)
       qed simp
-      ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s') (Th th1) = {}" by auto
+      ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s) (Th th1) = {}" by auto
       thus ?thesis by simp
      qed
   qed
   moreover have "subtree ?RAG'' (Th th1) =  subtree ?RAG' (Th th1)"
   proof(rule subtree_insert_next)
-    show "Th th' \<notin> subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)"
+    show "Th taker \<notin> subtree (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) (Th th1)"
     proof(rule subtree_refute)
-      show "Th th1 \<notin> ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')"
+      show "Th th1 \<notin> ancestors (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) (Th taker)"
             (is "_ \<notin> ?R")
       proof -
-          have "?R \<subseteq> ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto)
+          have "?R \<subseteq> ancestors (RAG s) (Th taker)" by (rule ancestors_mono, auto)
           moreover have "Th th1 \<notin> ..." using ancestors_th' assms by simp
           ultimately show ?thesis by auto
       qed
     next
-      from assms show "Th th1 \<noteq> Th th'" by simp
+      from assms show "Th th1 \<noteq> Th taker" by simp
     qed
   qed
   ultimately show ?thesis by (unfold RAG_s, simp)
 qed
 
 lemma cp_kept:
-  assumes "th1 \<notin> {th, th'}"
-  shows "cp s th1 = cp s' th1"
-    by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+  assumes "th1 \<notin> {th, taker}"
+  shows "cp (e#s) th1 = cp s th1"
+    by (unfold cp_alt_def the_preced_es subtree_kept[OF assms], simp)
 
 end
 
-locale step_v_cps_nnt = step_v_cps +
-  assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
 
-context step_v_cps_nnt
+context valid_trace_v_e
 begin
 
-lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"
-proof -
-  from nnt and  step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
-  show ?thesis by auto
-qed
+find_theorems RAG s e
+
+lemma RAG_s: "RAG (e#s) = RAG s - {(Cs cs, Th th)}"
+  by (unfold RAG_es waiting_set_eq holding_set_eq, simp)
 
 lemma subtree_kept:
   assumes "th1 \<noteq> th"
-  shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)"
+  shows "subtree (RAG (e#s)) (Th th1) = subtree (RAG s) (Th th1)"
 proof(unfold RAG_s, rule subset_del_subtree_outside)
-  show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s') (Th th1) = {}"
+  show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s) (Th th1) = {}"
   proof -
-    have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+    have "(Th th) \<notin> subtree (RAG s) (Th th1)"
     proof(rule subtree_refute)
-      show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+      show "Th th1 \<notin> ancestors (RAG s) (Th th)"
           by (unfold ancestors_th, simp)
     next
       from assms show "Th th1 \<noteq> Th th" by simp
@@ -387,94 +264,72 @@
 
 lemma cp_kept_1:
   assumes "th1 \<noteq> th"
-  shows "cp s th1 = cp s' th1"
-    by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+  shows "cp (e#s) th1 = cp s th1"
+    by (unfold cp_alt_def the_preced_es subtree_kept[OF assms], simp)
 
-lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}"
+lemma subtree_cs: "subtree (RAG s) (Cs cs) = {Cs cs}"
 proof -
   { fix n
-    have "(Cs cs) \<notin> ancestors (RAG s') n"
+    have "(Cs cs) \<notin> ancestors (RAG s) n"
     proof
-      assume "Cs cs \<in> ancestors (RAG s') n"
-      hence "(n, Cs cs) \<in> (RAG s')^+" by (auto simp:ancestors_def)
-      from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s'" by auto
+      assume "Cs cs \<in> ancestors (RAG s) n"
+      hence "(n, Cs cs) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+      from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s" by auto
       then obtain th' where "nn = Th th'"
         by (unfold s_RAG_def, auto)
-      from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s'" .
+      from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s" .
       from this[unfolded s_RAG_def]
-      have "waiting (wq s') th' cs" by auto
+      have "waiting (wq s) th' cs" by auto
       from this[unfolded cs_waiting_def]
-      have "1 < length (wq s' cs)"
-          by (cases "wq s' cs", auto)
-      from holding_next_thI[OF holding_th this]
-      obtain th' where "next_th s' th cs th'" by auto
-      with nnt show False by auto
+      have "1 < length (wq s cs)"
+          by (cases "wq s cs", auto)
+      from holding_next_thI[OF holding_th_cs_s this]
+      obtain th' where "next_th s th cs th'" by auto
+      thus False using no_taker by blast
     qed
   } note h = this
   {  fix n
-     assume "n \<in> subtree (RAG s') (Cs cs)"
+     assume "n \<in> subtree (RAG s) (Cs cs)"
      hence "n = (Cs cs)"
      by (elim subtreeE, insert h, auto)
-  } moreover have "(Cs cs) \<in> subtree (RAG s') (Cs cs)"
+  } moreover have "(Cs cs) \<in> subtree (RAG s) (Cs cs)"
       by (auto simp:subtree_def)
   ultimately show ?thesis by auto 
 qed
 
 lemma subtree_th: 
-  "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
-proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
+  "subtree (RAG (e#s)) (Th th) = subtree (RAG s) (Th th) - {Cs cs}"
+proof(unfold RAG_s, fold subtree_cs, rule rtree_RAG.subtree_del_inside)
   from edge_of_th
-  show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"
+  show "(Cs cs, Th th) \<in> edges_in (RAG s) (Th th)"
     by (unfold edges_in_def, auto simp:subtree_def)
 qed
 
 lemma cp_kept_2: 
-  shows "cp s th = cp s' th" 
- by (unfold cp_alt_def subtree_th preced_kept, auto)
+  shows "cp (e#s) th = cp s th" 
+ by (unfold cp_alt_def subtree_th the_preced_es, auto)
 
 lemma eq_cp:
-  shows "cp s th' = cp s' th'"
+  shows "cp (e#s) th' = cp s th'"
   using cp_kept_1 cp_kept_2
   by (cases "th' = th", auto)
+
 end
 
 
-locale step_P_cps =
-  fixes s' th cs s 
-  defines s_def : "s \<equiv> (P th cs#s')"
-  assumes vt_s: "vt s"
-
-sublocale step_P_cps < vat_s : valid_trace "s"
-proof
-  from vt_s show "vt s" .
-qed
-
 section {* The @{term P} operation *}
 
-sublocale step_P_cps < vat_s' : valid_trace "s'"
-proof
-  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_P_cps
+context valid_trace_p
 begin
 
-lemma readys_th: "th \<in> readys s'"
-proof -
-    from step_back_step [OF vt_s[unfolded s_def]]
-    have "PIP s' (P th cs)" .
-    hence "th \<in> runing s'" by (cases, simp)
-    thus ?thesis by (simp add:readys_def runing_def)
-qed
-
-lemma root_th: "root (RAG s') (Th th)"
-  using readys_root[OF readys_th] .
+lemma root_th: "root (RAG s) (Th th)"
+  by (simp add: ready_th_s readys_root)
 
 lemma in_no_others_subtree:
   assumes "th' \<noteq> th"
-  shows "Th th \<notin> subtree (RAG s') (Th th')"
+  shows "Th th \<notin> subtree (RAG s) (Th th')"
 proof
-  assume "Th th \<in> subtree (RAG s') (Th th')"
+  assume "Th th \<in> subtree (RAG s) (Th th')"
   thus False
   proof(cases rule:subtreeE)
     case 1
@@ -485,162 +340,140 @@
   qed
 qed
 
-lemma preced_kept: "the_preced s = the_preced s'"
-  by (auto simp: s_def the_preced_def preced_def)
+lemma preced_kept: "the_preced (e#s) = the_preced s"
+proof
+  fix th'
+  show "the_preced (e # s) th' = the_preced s th'"
+    by (unfold the_preced_def is_p preced_def, simp)
+qed
 
 end
 
-locale step_P_cps_ne =step_P_cps +
-  fixes th'
-  assumes ne: "wq s' cs \<noteq> []"
-  defines th'_def: "th' \<equiv> hd (wq s' cs)"
 
-locale step_P_cps_e =step_P_cps +
-  assumes ee: "wq s' cs = []"
-
-context step_P_cps_e
+context valid_trace_p_h
 begin
 
-lemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"
-proof -
-  from ee and  step_RAG_p[OF vt_s[unfolded s_def], folded s_def]
-  show ?thesis by auto
-qed
-
 lemma subtree_kept:
   assumes "th' \<noteq> th"
-  shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')"
-proof(unfold RAG_s, rule subtree_insert_next)
+  shows "subtree (RAG (e#s)) (Th th') = subtree (RAG s) (Th th')"
+proof(unfold RAG_es, rule subtree_insert_next)
   from in_no_others_subtree[OF assms] 
-  show "Th th \<notin> subtree (RAG s') (Th th')" .
+  show "Th th \<notin> subtree (RAG s) (Th th')" .
 qed
 
 lemma cp_kept: 
   assumes "th' \<noteq> th"
-  shows "cp s th' = cp s' th'"
+  shows "cp (e#s) th' = cp s th'"
 proof -
-  have "(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
-        (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+  have "(the_preced (e#s) ` {th'a. Th th'a \<in> subtree (RAG (e#s)) (Th th')}) =
+        (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')})"
         by (unfold preced_kept subtree_kept[OF assms], simp)
   thus ?thesis by (unfold cp_alt_def, simp)
 qed
 
 end
 
-context step_P_cps_ne 
+context valid_trace_p_w
 begin
 
-lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
-proof -
-  from step_RAG_p[OF vt_s[unfolded s_def]] and ne
-  show ?thesis by (simp add:s_def)
-qed
+interpretation vat_e: valid_trace "e#s"
+  by (unfold_locales, insert vt_e, simp)
 
-lemma cs_held: "(Cs cs, Th th') \<in> RAG s'"
-proof -
-  have "(Cs cs, Th th') \<in> hRAG s'"
-  proof -
-    from ne
-    have " holding s' th' cs"
-      by (unfold th'_def holding_eq cs_holding_def, auto)
-    thus ?thesis 
-      by (unfold hRAG_def, auto)
-  qed
-  thus ?thesis by (unfold RAG_split, auto)
-qed
+lemma cs_held: "(Cs cs, Th holder) \<in> RAG s"
+  using holding_s_holder
+  by (unfold s_RAG_def, fold holding_eq, auto)
 
 lemma tRAG_s: 
-  "tRAG s = tRAG s' \<union> {(Th th, Th th')}"
-  using RAG_tRAG_transfer[OF RAG_s cs_held] .
+  "tRAG (e#s) = tRAG s \<union> {(Th th, Th holder)}"
+  using local.RAG_tRAG_transfer[OF RAG_es cs_held] .
 
 lemma cp_kept:
-  assumes "Th th'' \<notin> ancestors (tRAG s) (Th th)"
-  shows "cp s th'' = cp s' th''"
+  assumes "Th th'' \<notin> ancestors (tRAG (e#s)) (Th th)"
+  shows "cp (e#s) th'' = cp s th''"
 proof -
-  have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')"
+  have h: "subtree (tRAG (e#s)) (Th th'') = subtree (tRAG s) (Th th'')"
   proof -
-    have "Th th' \<notin> subtree (tRAG s') (Th th'')"
+    have "Th holder \<notin> subtree (tRAG s) (Th th'')"
     proof
-      assume "Th th' \<in> subtree (tRAG s') (Th th'')"
+      assume "Th holder \<in> subtree (tRAG s) (Th th'')"
       thus False
       proof(rule subtreeE)
-         assume "Th th' = Th th''"
+         assume "Th holder = Th th''"
          from assms[unfolded tRAG_s ancestors_def, folded this]
          show ?thesis by auto
       next
-         assume "Th th'' \<in> ancestors (tRAG s') (Th th')"
-         moreover have "... \<subseteq> ancestors (tRAG s) (Th th')"
+         assume "Th th'' \<in> ancestors (tRAG s) (Th holder)"
+         moreover have "... \<subseteq> ancestors (tRAG (e#s)) (Th holder)"
          proof(rule ancestors_mono)
-            show "tRAG s' \<subseteq> tRAG s" by (unfold tRAG_s, auto)
+            show "tRAG s \<subseteq> tRAG (e#s)" by (unfold tRAG_s, auto)
          qed 
-         ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th')" by auto
-         moreover have "Th th' \<in> ancestors (tRAG s) (Th th)"
+         ultimately have "Th th'' \<in> ancestors (tRAG (e#s)) (Th holder)" by auto
+         moreover have "Th holder \<in> ancestors (tRAG (e#s)) (Th th)"
            by (unfold tRAG_s, auto simp:ancestors_def)
-         ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th)"
+         ultimately have "Th th'' \<in> ancestors (tRAG (e#s)) (Th th)"
                        by (auto simp:ancestors_def)
          with assms show ?thesis by auto
       qed
     qed
     from subtree_insert_next[OF this]
-    have "subtree (tRAG s' \<union> {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" .
+    have "subtree (tRAG s \<union> {(Th th, Th holder)}) (Th th'') = subtree (tRAG s) (Th th'')" .
     from this[folded tRAG_s] show ?thesis .
   qed
   show ?thesis by (unfold cp_alt_def1 h preced_kept, simp)
 qed
 
 lemma cp_gen_update_stop: (* ddd *)
-  assumes "u \<in> ancestors (tRAG s) (Th th)"
-  and "cp_gen s u = cp_gen s' u"
-  and "y \<in> ancestors (tRAG s) u"
-  shows "cp_gen s y = cp_gen s' y"
+  assumes "u \<in> ancestors (tRAG (e#s)) (Th th)"
+  and "cp_gen (e#s) u = cp_gen s u"
+  and "y \<in> ancestors (tRAG (e#s)) u"
+  shows "cp_gen (e#s) y = cp_gen s y"
   using assms(3)
-proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf])
+proof(induct rule:wf_induct[OF vat_e.fsbttRAGs.wf])
   case (1 x)
   show ?case (is "?L = ?R")
   proof -
     from tRAG_ancestorsE[OF 1(2)]
     obtain th2 where eq_x: "x = Th th2" by blast
-    from vat_s.cp_gen_rec[OF this]
+    from vat_e.cp_gen_rec[OF this]
     have "?L = 
-          Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)" .
+          Max ({the_preced (e#s) th2} \<union> cp_gen (e#s) ` RTree.children (tRAG (e#s)) x)" .
     also have "... = 
-          Max ({the_preced s' th2} \<union> cp_gen s' ` RTree.children (tRAG s') x)"
-  
+          Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)"
     proof -
-      from preced_kept have "the_preced s th2 = the_preced s' th2" by simp
-      moreover have "cp_gen s ` RTree.children (tRAG s) x =
-                     cp_gen s' ` RTree.children (tRAG s') x"
+      from preced_kept have "the_preced (e#s) th2 = the_preced s th2" by simp
+      moreover have "cp_gen (e#s) ` RTree.children (tRAG (e#s)) x =
+                     cp_gen s ` RTree.children (tRAG s) x"
       proof -
-        have "RTree.children (tRAG s) x =  RTree.children (tRAG s') x"
+        have "RTree.children (tRAG (e#s)) x =  RTree.children (tRAG s) x"
         proof(unfold tRAG_s, rule children_union_kept)
-          have start: "(Th th, Th th') \<in> tRAG s"
+          have start: "(Th th, Th holder) \<in> tRAG (e#s)"
             by (unfold tRAG_s, auto)
           note x_u = 1(2)
-          show "x \<notin> Range {(Th th, Th th')}"
+          show "x \<notin> Range {(Th th, Th holder)}"
           proof
-            assume "x \<in> Range {(Th th, Th th')}"
-            hence eq_x: "x = Th th'" using RangeE by auto
+            assume "x \<in> Range {(Th th, Th holder)}"
+            hence eq_x: "x = Th holder" using RangeE by auto
             show False
-            proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start])
+            proof(cases rule:vat_e.ancestors_headE[OF assms(1) start])
               case 1
-              from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG
+              from x_u[folded this, unfolded eq_x] vat_e.acyclic_tRAG
               show ?thesis by (auto simp:ancestors_def acyclic_def)
             next
               case 2
               with x_u[unfolded eq_x]
-              have "(Th th', Th th') \<in> (tRAG s)^+" by (auto simp:ancestors_def)
-              with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
+              have "(Th holder, Th holder) \<in> (tRAG (e#s))^+" by (auto simp:ancestors_def)
+              with vat_e.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
             qed
           qed
         qed
-        moreover have "cp_gen s ` RTree.children (tRAG s) x =
-                       cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A")
+        moreover have "cp_gen (e#s) ` RTree.children (tRAG (e#s)) x =
+                       cp_gen s ` RTree.children (tRAG (e#s)) x" (is "?f ` ?A = ?g ` ?A")
         proof(rule f_image_eq)
           fix a
           assume a_in: "a \<in> ?A"
           from 1(2)
           show "?f a = ?g a"
-          proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
+          proof(cases rule:vat_e.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
              case in_ch
              show ?thesis
              proof(cases "a = u")
@@ -648,58 +481,58 @@
                 from assms(2)[folded this] show ?thesis .
              next
                 case False
-                have a_not_in: "a \<notin> ancestors (tRAG s) (Th th)"
+                have a_not_in: "a \<notin> ancestors (tRAG (e#s)) (Th th)"
                 proof
-                  assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+                  assume a_in': "a \<in> ancestors (tRAG (e#s)) (Th th)"
                   have "a = u"
-                  proof(rule vat_s.rtree_s.ancestors_children_unique)
-                    from a_in' a_in show "a \<in> ancestors (tRAG s) (Th th) \<inter> 
-                                          RTree.children (tRAG s) x" by auto
+                  proof(rule vat_e.rtree_s.ancestors_children_unique)
+                    from a_in' a_in show "a \<in> ancestors (tRAG (e#s)) (Th th) \<inter> 
+                                          RTree.children (tRAG (e#s)) x" by auto
                   next 
-                    from assms(1) in_ch show "u \<in> ancestors (tRAG s) (Th th) \<inter> 
-                                      RTree.children (tRAG s) x" by auto
+                    from assms(1) in_ch show "u \<in> ancestors (tRAG (e#s)) (Th th) \<inter> 
+                                      RTree.children (tRAG (e#s)) x" by auto
                   qed
                   with False show False by simp
                 qed
                 from a_in obtain th_a where eq_a: "a = Th th_a" 
                     by (unfold RTree.children_def tRAG_alt_def, auto)
                 from cp_kept[OF a_not_in[unfolded eq_a]]
-                have "cp s th_a = cp s' th_a" .
+                have "cp (e#s) th_a = cp s th_a" .
                 from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
                 show ?thesis .
              qed
           next
             case (out_ch z)
-            hence h: "z \<in> ancestors (tRAG s) u" "z \<in> RTree.children (tRAG s) x" by auto
+            hence h: "z \<in> ancestors (tRAG (e#s)) u" "z \<in> RTree.children (tRAG (e#s)) x" by auto
             show ?thesis
             proof(cases "a = z")
               case True
-              from h(2) have zx_in: "(z, x) \<in> (tRAG s)" by (auto simp:RTree.children_def)
+              from h(2) have zx_in: "(z, x) \<in> (tRAG (e#s))" by (auto simp:RTree.children_def)
               from 1(1)[rule_format, OF this h(1)]
-              have eq_cp_gen: "cp_gen s z = cp_gen s' z" .
+              have eq_cp_gen: "cp_gen (e#s) z = cp_gen s z" .
               with True show ?thesis by metis
             next
               case False
               from a_in obtain th_a where eq_a: "a = Th th_a"
                 by (auto simp:RTree.children_def tRAG_alt_def)
-              have "a \<notin> ancestors (tRAG s) (Th th)"
+              have "a \<notin> ancestors (tRAG (e#s)) (Th th)"
               proof
-                assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+                assume a_in': "a \<in> ancestors (tRAG (e#s)) (Th th)"
                 have "a = z"
-                proof(rule vat_s.rtree_s.ancestors_children_unique)
-                  from assms(1) h(1) have "z \<in> ancestors (tRAG s) (Th th)"
+                proof(rule vat_e.rtree_s.ancestors_children_unique)
+                  from assms(1) h(1) have "z \<in> ancestors (tRAG (e#s)) (Th th)"
                       by (auto simp:ancestors_def)
-                  with h(2) show " z \<in> ancestors (tRAG s) (Th th) \<inter> 
-                                       RTree.children (tRAG s) x" by auto
+                  with h(2) show " z \<in> ancestors (tRAG (e#s)) (Th th) \<inter> 
+                                       RTree.children (tRAG (e#s)) x" by auto
                 next
                   from a_in a_in'
-                  show "a \<in> ancestors (tRAG s) (Th th) \<inter> RTree.children (tRAG s) x"
+                  show "a \<in> ancestors (tRAG (e#s)) (Th th) \<inter> RTree.children (tRAG (e#s)) x"
                     by auto
                 qed
                 with False show False by auto
               qed
               from cp_kept[OF this[unfolded eq_a]]
-              have "cp s th_a = cp s' th_a" .
+              have "cp (e#s) th_a = cp s th_a" .
               from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
               show ?thesis .
             qed
@@ -710,21 +543,21 @@
       ultimately show ?thesis by simp
     qed
     also have "... = ?R"
-      by (fold vat_s'.cp_gen_rec[OF eq_x], simp)
+      by (fold cp_gen_rec[OF eq_x], simp)
     finally show ?thesis .
   qed
 qed
 
 lemma cp_up:
-  assumes "(Th th') \<in> ancestors (tRAG s) (Th th)"
-  and "cp s th' = cp s' th'"
-  and "(Th th'') \<in> ancestors (tRAG s) (Th th')"
-  shows "cp s th'' = cp s' th''"
+  assumes "(Th th') \<in> ancestors (tRAG (e#s)) (Th th)"
+  and "cp (e#s) th' = cp s th'"
+  and "(Th th'') \<in> ancestors (tRAG (e#s)) (Th th')"
+  shows "cp (e#s) th'' = cp s th''"
 proof -
-  have "cp_gen s (Th th'') = cp_gen s' (Th th'')"
+  have "cp_gen (e#s) (Th th'') = cp_gen s (Th th'')"
   proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)])
     from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]]
-    show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis
+    show "cp_gen (e#s) (Th th') = cp_gen s (Th th')" by metis
   qed
   with cp_gen_def_cond[OF refl[of "Th th''"]]
   show ?thesis by metis
@@ -734,50 +567,32 @@
 
 section {* The @{term Create} operation *}
 
-locale step_create_cps =
-  fixes s' th prio s 
-  defines s_def : "s \<equiv> (Create th prio#s')"
-  assumes vt_s: "vt s"
-
-sublocale step_create_cps < vat_s: valid_trace "s"
-  by (unfold_locales, insert vt_s, simp)
+context valid_trace_create
+begin 
 
-sublocale step_create_cps < vat_s': valid_trace "s'"
-  by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
-
-context step_create_cps
-begin
+interpretation vat_e: valid_trace "e#s"
+  by (unfold_locales, insert vt_e, simp)
 
-lemma RAG_kept: "RAG s = RAG s'"
-  by (unfold s_def RAG_create_unchanged, auto)
-
-lemma tRAG_kept: "tRAG s = tRAG s'"
-  by (unfold tRAG_alt_def RAG_kept, auto)
+lemma tRAG_kept: "tRAG (e#s) = tRAG s"
+  by (unfold tRAG_alt_def RAG_unchanged, auto)
 
 lemma preced_kept:
   assumes "th' \<noteq> th"
-  shows "the_preced s th' = the_preced s' th'"
-  by (unfold s_def the_preced_def preced_def, insert assms, auto)
+  shows "the_preced (e#s) th' = the_preced s th'"
+  by (unfold the_preced_def preced_def is_create, insert assms, auto)
 
-lemma th_not_in: "Th th \<notin> Field (tRAG s')"
-proof -
-  from vt_s[unfolded s_def]
-  have "PIP s' (Create th prio)" by (cases, simp)
-  hence "th \<notin> threads s'" by(cases, simp)
-  from vat_s'.not_in_thread_isolated[OF this]
-  have "Th th \<notin> Field (RAG s')" .
-  with tRAG_Field show ?thesis by auto
-qed
+lemma th_not_in: "Th th \<notin> Field (tRAG s)"
+  by (meson not_in_thread_isolated subsetCE tRAG_Field th_not_live_s)
 
 lemma eq_cp:
   assumes neq_th: "th' \<noteq> th"
-  shows "cp s th' = cp s' th'"
+  shows "cp (e#s) th' = cp s th'"
 proof -
-  have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
-        (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+  have "(the_preced (e#s) \<circ> the_thread) ` subtree (tRAG (e#s)) (Th th') =
+        (the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th')"
   proof(unfold tRAG_kept, rule f_image_eq)
     fix a
-    assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+    assume a_in: "a \<in> subtree (tRAG s) (Th th')"
     then obtain th_a where eq_a: "a = Th th_a" 
     proof(cases rule:subtreeE)
       case 2
@@ -786,9 +601,9 @@
     qed auto
     have neq_th_a: "th_a \<noteq> th"
     proof -
-      have "(Th th) \<notin> subtree (tRAG s') (Th th')"
+      have "(Th th) \<notin> subtree (tRAG s) (Th th')"
       proof
-        assume "Th th \<in> subtree (tRAG s') (Th th')"
+        assume "Th th \<in> subtree (tRAG s) (Th th')"
         thus False
         proof(cases rule:subtreeE)
           case 2
@@ -800,99 +615,72 @@
       with a_in[unfolded eq_a] show ?thesis by auto
     qed
     from preced_kept[OF this]
-    show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+    show "(the_preced (e#s) \<circ> the_thread) a = (the_preced s \<circ> the_thread) a"
       by (unfold eq_a, simp)
   qed
   thus ?thesis by (unfold cp_alt_def1, simp)
 qed
 
-lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}"
+lemma children_of_th: "RTree.children (tRAG (e#s)) (Th th) = {}"
 proof -
   { fix a
-    assume "a \<in> RTree.children (tRAG s) (Th th)"
-    hence "(a, Th th) \<in> tRAG s" by (auto simp:RTree.children_def)
+    assume "a \<in> RTree.children (tRAG (e#s)) (Th th)"
+    hence "(a, Th th) \<in> tRAG (e#s)" by (auto simp:RTree.children_def)
     with th_not_in have False 
      by (unfold Field_def tRAG_kept, auto)
   } thus ?thesis by auto
 qed
 
-lemma eq_cp_th: "cp s th = preced th s"
- by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def)
+lemma eq_cp_th: "cp (e#s) th = preced th (e#s)"
+ by (unfold vat_e.cp_rec children_of_th, simp add:the_preced_def)
 
 end
 
-locale step_exit_cps =
-  fixes s' th prio s 
-  defines s_def : "s \<equiv> Exit th # s'"
-  assumes vt_s: "vt s"
 
-sublocale step_exit_cps < vat_s: valid_trace "s"
-  by (unfold_locales, insert vt_s, simp)
-
-sublocale step_exit_cps < vat_s': valid_trace "s'"
-  by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
-
-context step_exit_cps
+context valid_trace_exit
 begin
 
 lemma preced_kept:
   assumes "th' \<noteq> th"
-  shows "the_preced s th' = the_preced s' th'"
-  by (unfold s_def the_preced_def preced_def, insert assms, auto)
-
-lemma RAG_kept: "RAG s = RAG s'"
-  by (unfold s_def RAG_exit_unchanged, auto)
-
-lemma tRAG_kept: "tRAG s = tRAG s'"
-  by (unfold tRAG_alt_def RAG_kept, auto)
+  shows "the_preced (e#s) th' = the_preced s th'"
+  using assms
+  by (unfold the_preced_def is_exit preced_def, simp)
 
-lemma th_ready: "th \<in> readys s'"
-proof -
-  from vt_s[unfolded s_def]
-  have "PIP s' (Exit th)" by (cases, simp)
-  hence h: "th \<in> runing s' \<and> holdents s' th = {}" by (cases, metis)
-  thus ?thesis by (unfold runing_def, auto)
-qed
+lemma tRAG_kept: "tRAG (e#s) = tRAG s"
+  by (unfold tRAG_alt_def RAG_unchanged, auto)
 
-lemma th_holdents: "holdents s' th = {}"
+lemma th_RAG: "Th th \<notin> Field (RAG s)"
 proof -
- from vt_s[unfolded s_def]
-  have "PIP s' (Exit th)" by (cases, simp)
-  thus ?thesis by (cases, metis)
-qed
-
-lemma th_RAG: "Th th \<notin> Field (RAG s')"
-proof -
-  have "Th th \<notin> Range (RAG s')"
+  have "Th th \<notin> Range (RAG s)"
   proof
-    assume "Th th \<in> Range (RAG s')"
-    then obtain cs where "holding (wq s') th cs"
+    assume "Th th \<in> Range (RAG s)"
+    then obtain cs where "holding (wq s) th cs"
       by (unfold Range_iff s_RAG_def, auto)
-    with th_holdents[unfolded holdents_def]
-    show False by (unfold eq_holding, auto)
+    with holdents_th_s[unfolded holdents_def]
+    show False by (unfold holding_eq, auto)
   qed
-  moreover have "Th th \<notin> Domain (RAG s')"
+  moreover have "Th th \<notin> Domain (RAG s)"
   proof
-    assume "Th th \<in> Domain (RAG s')"
-    then obtain cs where "waiting (wq s') th cs"
+    assume "Th th \<in> Domain (RAG s)"
+    then obtain cs where "waiting (wq s) th cs"
       by (unfold Domain_iff s_RAG_def, auto)
-    with th_ready show False by (unfold readys_def eq_waiting, auto)
+    with th_ready_s show False by (unfold readys_def waiting_eq, auto)
   qed
   ultimately show ?thesis by (auto simp:Field_def)
 qed
 
-lemma th_tRAG: "(Th th) \<notin> Field (tRAG s')"
-  using th_RAG tRAG_Field[of s'] by auto
+lemma th_tRAG: "(Th th) \<notin> Field (tRAG s)"
+  using th_RAG tRAG_Field by auto
 
 lemma eq_cp:
   assumes neq_th: "th' \<noteq> th"
-  shows "cp s th' = cp s' th'"
+  shows "cp (e#s) th' = cp s th'"
 proof -
-  have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
-        (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+  have "(the_preced (e#s) \<circ> the_thread) ` subtree (tRAG (e#s)) (Th th') =
+        (the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th')"
   proof(unfold tRAG_kept, rule f_image_eq)
     fix a
-    assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+    assume a_in: "a \<in> subtree (tRAG s) (Th th')"
     then obtain th_a where eq_a: "a = Th th_a" 
     proof(cases rule:subtreeE)
       case 2
@@ -901,14 +689,14 @@
     qed auto
     have neq_th_a: "th_a \<noteq> th"
     proof -
-      from vat_s'.readys_in_no_subtree[OF th_ready assms]
-      have "(Th th) \<notin> subtree (RAG s') (Th th')" .
-      with tRAG_subtree_RAG[of s' "Th th'"]
-      have "(Th th) \<notin> subtree (tRAG s') (Th th')" by auto
+      from readys_in_no_subtree[OF th_ready_s assms]
+      have "(Th th) \<notin> subtree (RAG s) (Th th')" .
+      with tRAG_subtree_RAG[of s "Th th'"]
+      have "(Th th) \<notin> subtree (tRAG s) (Th th')" by auto
       with a_in[unfolded eq_a] show ?thesis by auto
     qed
     from preced_kept[OF this]
-    show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+    show "(the_preced (e#s) \<circ> the_thread) a = (the_preced s \<circ> the_thread) a"
       by (unfold eq_a, simp)
   qed
   thus ?thesis by (unfold cp_alt_def1, simp)
@@ -918,3 +706,924 @@
 
 end
 
+=======
+theory ExtGG
+imports PrioG CpsG
+begin
+
+text {* 
+  The following two auxiliary lemmas are used to reason about @{term Max}.
+*}
+lemma image_Max_eqI: 
+  assumes "finite B"
+  and "b \<in> B"
+  and "\<forall> x \<in> B. f x \<le> f b"
+  shows "Max (f ` B) = f b"
+  using assms
+  using Max_eqI by blast 
+
+lemma image_Max_subset:
+  assumes "finite A"
+  and "B \<subseteq> A"
+  and "a \<in> B"
+  and "Max (f ` A) = f a"
+  shows "Max (f ` B) = f a"
+proof(rule image_Max_eqI)
+  show "finite B"
+    using assms(1) assms(2) finite_subset by auto 
+next
+  show "a \<in> B" using assms by simp
+next
+  show "\<forall>x\<in>B. f x \<le> f a"
+    by (metis Max_ge assms(1) assms(2) assms(4) 
+            finite_imageI image_eqI subsetCE) 
+qed
+
+text {*
+  The following locale @{text "highest_gen"} sets the basic context for our
+  investigation: supposing thread @{text th} holds the highest @{term cp}-value
+  in state @{text s}, which means the task for @{text th} is the 
+  most urgent. We want to show that  
+  @{text th} is treated correctly by PIP, which means
+  @{text th} will not be blocked unreasonably by other less urgent
+  threads. 
+*}
+locale highest_gen =
+  fixes s th prio tm
+  assumes vt_s: "vt s"
+  and threads_s: "th \<in> threads s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+  -- {* The internal structure of @{term th}'s precedence is exposed:*}
+  and preced_th: "preced th s = Prc prio tm" 
+
+-- {* @{term s} is a valid trace, so it will inherit all results derived for
+      a valid trace: *}
+sublocale highest_gen < vat_s: valid_trace "s"
+  by (unfold_locales, insert vt_s, simp)
+
+context highest_gen
+begin
+
+text {*
+  @{term tm} is the time when the precedence of @{term th} is set, so 
+  @{term tm} must be a valid moment index into @{term s}.
+*}
+lemma lt_tm: "tm < length s"
+  by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+text {*
+  Since @{term th} holds the highest precedence and @{text "cp"}
+  is the highest precedence of all threads in the sub-tree of 
+  @{text "th"} and @{text th} is among these threads, 
+  its @{term cp} must equal to its precedence:
+*}
+lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
+proof -
+  have "?L \<le> ?R"
+  by (unfold highest, rule Max_ge, 
+        auto simp:threads_s finite_threads)
+  moreover have "?R \<le> ?L"
+    by (unfold vat_s.cp_rec, rule Max_ge, 
+        auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
+  ultimately show ?thesis by auto
+qed
+
+(* ccc *)
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq, unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[symmetric]
+  show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen + 
+  fixes t 
+  assumes vt_t: "vt (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+sublocale extend_highest_gen < vat_t: valid_trace "t@s"
+  by (unfold_locales, insert vt_t, simp)
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt (t@s)" 
+  shows "vt s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt (t @ s) \<Longrightarrow> vt s"
+      and vt_et: "vt ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+
+locale red_extend_highest_gen = extend_highest_gen +
+   fixes i::nat
+
+sublocale red_extend_highest_gen <   red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
+  apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+  by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+
+context extend_highest_gen
+begin
+
+ lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e; 
+                    extend_highest_gen s th prio tm t; 
+                    extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_gen_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt ((e # t') @ s)"
+      and et: "extend_highest_gen s th prio tm (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_gen s th prio tm (e # t')" .
+    next
+      from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+                 preced th (t@s) = preced th s" (is "?Q t") 
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show ?case
+      by auto
+  next
+    case (Cons e t)
+    interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+    interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      show ?thesis
+      proof -
+        from Cons and Create have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          case thread_create
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold Create, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:Create)
+      qed
+    next
+      case (Exit thread)
+      from h_e.exit_diff and Exit
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold Exit, auto simp:preced_def)
+    next
+      case (P thread cs)
+      with Cons
+      show ?thesis 
+        by (auto simp:P preced_def)
+    next
+      case (V thread cs)
+      with Cons
+      show ?thesis 
+        by (auto simp:V preced_def)
+    next
+      case (Set thread prio')
+      show ?thesis
+      proof -
+        from h_e.set_diff_low and Set
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold Set, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:Set)
+      qed
+    qed
+  qed
+qed
+
+text {*
+  According to @{thm th_kept}, thread @{text "th"} has its living status
+  and precedence kept along the way of @{text "t"}. The following lemma
+  shows that this preserved precedence of @{text "th"} remains as the highest
+  along the way of @{text "t"}.
+
+  The proof goes by induction over @{text "t"} using the specialized
+  induction rule @{thm ind}, followed by case analysis of each possible 
+  operations of PIP. All cases follow the same pattern rendered by the 
+  generalized introduction rule @{thm "image_Max_eqI"}. 
+
+  The very essence is to show that precedences, no matter whether they are newly introduced 
+  or modified, are always lower than the one held by @{term "th"},
+  which by @{thm th_kept} is preserved along the way.
+*}
+lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show ?case
+    by (unfold the_preced_def, simp)
+next
+  case (Cons e t)
+    interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+    interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      -- {* The following is the common pattern of each branch of the case analysis. *}
+      -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume "x \<in> ?A"
+          hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
+          thus "?f x \<le> ?f th"
+          proof
+            assume "x = thread"
+            thus ?thesis 
+              apply (simp add:Create the_preced_def preced_def, fold preced_def)
+              using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force
+          next
+            assume h: "x \<in> threads (t @ s)"
+            from Cons(2)[unfolded Create] 
+            have "x \<noteq> thread" using h by (cases, auto)
+            hence "?f x = the_preced (t@s) x" 
+              by (simp add:Create the_preced_def preced_def)
+            hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
+              by (simp add: h_t.finite_threads h)
+            also have "... = ?f th"
+              by (metis Cons.hyps(5) h_e.th_kept the_preced_def) 
+            finally show ?thesis .
+          qed
+        qed
+      qed
+     -- {* The minor part is to show that the precedence of @{text "th"} 
+           equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      -- {* Then it follows trivially that the precedence preserved
+            for @{term "th"} remains the maximum of all living threads along the way. *}
+      finally show ?thesis .
+    qed 
+  next 
+    case (Exit thread)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume "x \<in> ?A"
+          hence "x \<in> threads (t@s)" by (simp add: Exit) 
+          hence "?f x \<le> Max (?f ` threads (t@s))" 
+            by (simp add: h_t.finite_threads) 
+          also have "... \<le> ?f th" 
+            apply (simp add:Exit the_preced_def preced_def, fold preced_def)
+            using Cons.hyps(5) h_t.th_kept the_preced_def by auto
+          finally show "?f x \<le> ?f th" .
+        qed
+      qed
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      finally show ?thesis .
+    qed 
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def the_preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def the_preced_def)
+  next 
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume h: "x \<in> ?A"
+          show "?f x \<le> ?f th"
+          proof(cases "x = thread")
+            case True
+            moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
+            proof -
+              have "the_preced (t @ s) th = Prc prio tm"  
+                using h_t.th_kept preced_th by (simp add:the_preced_def)
+              moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
+              ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
+            qed
+            ultimately show ?thesis
+              by (unfold Set, simp add:the_preced_def preced_def)
+          next
+            case False
+            then have "?f x  = the_preced (t@s) x"
+              by (simp add:the_preced_def preced_def Set)
+            also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
+              using Set h h_t.finite_threads by auto 
+            also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) 
+            finally show ?thesis .
+          qed
+        qed
+      qed
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      finally show ?thesis .
+    qed 
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+text {*
+  The reason behind the following lemma is that:
+  Since @{term "cp"} is defined as the maximum precedence 
+  of those threads contained in the sub-tree of node @{term "Th th"} 
+  in @{term "RAG (t@s)"}, and all these threads are living threads, and 
+  @{term "th"} is also among them, the maximum precedence of 
+  them all must be the one for @{text "th"}.
+*}
+lemma th_cp_max_preced: 
+  "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") 
+proof -
+  let ?f = "the_preced (t@s)"
+  have "?L = ?f th"
+  proof(unfold cp_alt_def, rule image_Max_eqI)
+    show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+    proof -
+      have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} = 
+            the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
+                            (\<exists> th'. n = Th th')}"
+      by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
+      moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) 
+      ultimately show ?thesis by simp
+    qed
+  next
+    show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+      by (auto simp:subtree_def)
+  next
+    show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
+               the_preced (t @ s) x \<le> the_preced (t @ s) th"
+    proof
+      fix th'
+      assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+      hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
+      moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
+        by (meson subtree_Field)
+      ultimately have "Th th' \<in> ..." by auto
+      hence "th' \<in> threads (t@s)" 
+      proof
+        assume "Th th' \<in> {Th th}"
+        thus ?thesis using th_kept by auto 
+      next
+        assume "Th th' \<in> Field (RAG (t @ s))"
+        thus ?thesis using vat_t.not_in_thread_isolated by blast 
+      qed
+      thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
+        by (metis Max_ge finite_imageI finite_threads image_eqI 
+               max_kept th_kept the_preced_def)
+    qed
+  qed
+  also have "... = ?R" by (simp add: max_preced the_preced_def) 
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less:
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+  using assms
+by (metis Max.coboundedI finite_imageI highest not_le order.trans 
+    preced_linorder rev_image_eqI threads_s vat_s.finite_threads 
+    vat_s.le_cp)
+
+text {*
+  Counting of the number of @{term "P"} and @{term "V"} operations 
+  is the cornerstone of a large number of the following proofs. 
+  The reason is that this counting is quite easy to calculate and 
+  convenient to use in the reasoning. 
+
+  The following lemma shows that the counting controls whether 
+  a thread is running or not.
+*}
+
+lemma pv_blocked_pre:
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume otherwise: "th' \<in> runing (t@s)"
+  show False
+  proof -
+    have "th' = th"
+    proof(rule preced_unique)
+      show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+      proof -
+        have "?L = cp (t@s) th'"
+          by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
+        also have "... = cp (t @ s) th" using otherwise 
+          by (metis (mono_tags, lifting) mem_Collect_eq 
+                    runing_def th_cp_max vat_t.max_cp_readys_threads)
+        also have "... = ?R" by (metis th_cp_preced th_kept) 
+        finally show ?thesis .
+      qed
+    qed (auto simp: th'_in th_kept)
+    moreover have "th' \<noteq> th" using neq_th' .
+    ultimately show ?thesis by simp
+ qed
+qed
+
+lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof(induct rule:ind)
+  case (Cons e t)
+    interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
+    interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
+    show ?case
+    proof(cases e)
+      case (P thread cs)
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (P thread cs)" using Cons P by auto
+            thus ?thesis
+            proof(cases)
+              assume "thread \<in> runing (t@s)"
+              moreover have "th' \<notin> runing (t@s)" using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+              ultimately show ?thesis by auto
+            qed
+          qed with Cons show ?thesis
+            by (unfold P, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (V thread cs)
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (V thread cs)" using Cons V by auto
+            thus ?thesis
+            proof(cases)
+              assume "thread \<in> runing (t@s)"
+              moreover have "th' \<notin> runing (t@s)" using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+              ultimately show ?thesis by auto
+            qed
+          qed with Cons show ?thesis
+            by (unfold V, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Create thread prio')
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (Create thread prio')" using Cons Create by auto
+            thus ?thesis using Cons(5) by (cases, auto)
+          qed with Cons show ?thesis
+            by (unfold Create, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Exit thread)
+      show ?thesis
+      proof -
+        have neq_thread: "thread \<noteq> th'"
+        proof -
+          have "step (t@s) (Exit thread)" using Cons Exit by auto
+          thus ?thesis apply (cases) using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+        qed 
+        hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons
+            by (unfold Exit, simp add:cntP_def cntV_def count_def)
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread 
+          by (unfold Exit, simp) 
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+next
+  case Nil
+  with assms
+  show ?case by auto
+qed
+
+text {* Changing counting balance to detachedness *}
+lemmas runing_precond_pre_dtc = runing_precond_pre
+         [folded vat_t.detached_eq vat_s.detached_eq]
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+  using assms
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+    by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)
+  moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof -
+  interpret h_i: red_extend_highest_gen _ _ _ _ _ i
+      by (unfold_locales)
+  interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"
+      by (unfold_locales)
+  interpret h:  extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"
+  proof(unfold_locales)
+    show "vt (moment i t @ s)" by (metis h_i.vt_t) 
+  next
+    show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)
+  next
+    show "preced th (moment i t @ s) = 
+            Max (cp (moment i t @ s) ` threads (moment i t @ s))"
+              by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)
+  next
+    show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th) 
+  next
+    show "vt (moment j (restm i t) @ moment i t @ s)"
+      using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)
+  next
+    fix th' prio'
+    assume "Create th' prio' \<in> set (moment j (restm i t))"
+    thus "prio' \<le> prio" using assms
+       by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)
+  next
+    fix th' prio'
+    assume "Set th' prio' \<in> set (moment j (restm i t))"
+    thus "th' \<noteq> th \<and> prio' \<le> prio"
+    by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)
+  next
+    fix th'
+    assume "Exit th' \<in> set (moment j (restm i t))"
+    thus "th' \<noteq> th"
+      by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)
+  qed
+  show ?thesis 
+    by (metis add.commute append_assoc eq_pv h.runing_precond_pre
+          moment_plus_split neq_th' th'_in)
+qed
+
+lemma moment_blocked_eqpv:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+   and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  moreover have "th' \<notin> runing ((moment j t)@s)"
+  proof -
+    interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+    show ?thesis
+      using h.pv_blocked_pre h1 h2 neq_th' by auto 
+  qed
+  ultimately show ?thesis by auto
+qed
+
+(* The foregoing two lemmas are preparation for this one, but
+   in long run can be combined. Maybe I am wrong.
+*)
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and dtc: "detached (moment i t @ s) th'"
+  and le_ij: "i \<le> j"
+  shows "detached (moment j t @ s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+  interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales) 
+  have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"
+                by (metis dtc h_i.detached_elim)
+  from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]
+  show ?thesis by (metis h_j.detached_intro) 
+qed
+
+lemma runing_preced_inversion:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "cp (t@s) th' = preced th s" (is "?L = ?R")
+proof -
+  have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
+      by (unfold runing_def, auto)
+  also have "\<dots> = ?R"
+      by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) 
+  finally show ?thesis .
+qed
+
+text {*
+  The situation when @{term "th"} is blocked is analyzed by the following lemmas.
+*}
+
+text {*
+  The following lemmas shows the running thread @{text "th'"}, if it is different from
+  @{term th}, must be live at the very beginning. By the term {\em the very beginning},
+  we mean the moment where the formal investigation starts, i.e. the moment (or state)
+  @{term s}. 
+*}
+
+lemma runing_inversion_0:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s"
+proof -
+    -- {* The proof is by contradiction: *}
+    { assume otherwise: "\<not> ?thesis"
+      have "th' \<notin> runing (t @ s)"
+      proof -
+        -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}
+        have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)
+        -- {* However, @{text "th'"} does not exist at very beginning. *}
+        have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise
+          by (metis append.simps(1) moment_zero)
+        -- {* Therefore, there must be a moment during @{text "t"}, when 
+              @{text "th'"} came into being. *}
+        -- {* Let us suppose the moment being @{text "i"}: *}
+        from p_split_gen[OF th'_in th'_notin]
+        obtain i where lt_its: "i < length t"
+                 and le_i: "0 \<le> i"
+                 and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+                 and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by (auto)
+        interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+        interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)
+        from lt_its have "Suc i \<le> length t" by auto
+        -- {* Let us also suppose the event which makes this change is @{text e}: *}
+        from moment_head[OF this] obtain e where 
+          eq_me: "moment (Suc i) t = e # moment i t" by blast
+        hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t) 
+        hence "PIP (moment i t @ s) e" by (cases, simp)
+        -- {* It can be derived that this event @{text "e"}, which 
+              gives birth to @{term "th'"} must be a @{term "Create"}: *}
+        from create_pre[OF this, of th']
+        obtain prio where eq_e: "e = Create th' prio"
+            by (metis append_Cons eq_me lessI post pre) 
+        have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto 
+        have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+        proof -
+          have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+            by (metis h_i.cnp_cnv_eq pre)
+          thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)
+        qed
+        show ?thesis 
+          using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge
+            by auto
+      qed
+      with `th' \<in> runing (t@s)`
+      have False by simp
+    } thus ?thesis by auto
+qed
+
+text {* 
+  The second lemma says, if the running thread @{text th'} is different from 
+  @{term th}, then this @{text th'} must in the possession of some resources
+  at the very beginning. 
+
+  To ease the reasoning of resource possession of one particular thread, 
+  we used two auxiliary functions @{term cntV} and @{term cntP}, 
+  which are the counters of @{term P}-operations and 
+  @{term V}-operations respectively. 
+  If the number of @{term V}-operation is less than the number of 
+  @{term "P"}-operations, the thread must have some unreleased resource. 
+*}
+
+lemma runing_inversion_1: (* ddd *)
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  -- {* thread @{term "th'"} is a live on in state @{term "s"} and 
+        it has some unreleased resource. *}
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof -
+  -- {* The proof is a simple composition of @{thm runing_inversion_0} and 
+        @{thm runing_precond}: *}
+  -- {* By applying @{thm runing_inversion_0} to assumptions,
+        it can be shown that @{term th'} is live in state @{term s}: *}
+  have "th' \<in> threads s"  using runing_inversion_0[OF assms(1,2)] .
+  -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+qed
+
+text {* 
+  The following lemma is just a rephrasing of @{thm runing_inversion_1}:
+*}
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_3:
+  assumes runing': "th' \<in> runing (t@s)"
+  and neq_th: "th' \<noteq> th"
+  shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
+  by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
+
+lemma runing_inversion_4:
+  assumes runing': "th' \<in> runing (t@s)"
+  and neq_th: "th' \<noteq> th"
+  shows "th' \<in> threads s"
+  and    "\<not>detached s th'"
+  and    "cp (t@s) th' = preced th s"
+  apply (metis neq_th runing' runing_inversion_2)
+  apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
+  by (metis neq_th runing' runing_inversion_3)
+
+
+text {* 
+  Suppose @{term th} is not running, it is first shown that
+  there is a path in RAG leading from node @{term th} to another thread @{text "th'"} 
+  in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
+
+  Now, since @{term readys}-set is non-empty, there must be
+  one in it which holds the highest @{term cp}-value, which, by definition, 
+  is the @{term runing}-thread. However, we are going to show more: this running thread
+  is exactly @{term "th'"}.
+     *}
+lemma th_blockedE: (* ddd *)
+  assumes "th \<notin> runing (t@s)"
+  obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+                    "th' \<in> runing (t@s)"
+proof -
+  -- {* According to @{thm vat_t.th_chain_to_ready}, either 
+        @{term "th"} is in @{term "readys"} or there is path leading from it to 
+        one thread in @{term "readys"}. *}
+  have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)" 
+    using th_kept vat_t.th_chain_to_ready by auto
+  -- {* However, @{term th} can not be in @{term readys}, because otherwise, since 
+       @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
+  moreover have "th \<notin> readys (t@s)" 
+    using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto 
+  -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in 
+        term @{term readys}: *}
+  ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
+                          and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
+  -- {* We are going to show that this @{term th'} is running. *}
+  have "th' \<in> runing (t@s)"
+  proof -
+    -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
+    have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
+    proof -
+      have "?L =  Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
+        by (unfold cp_alt_def1, simp)
+      also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
+      proof(rule image_Max_subset)
+        show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
+      next
+        show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
+          by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) 
+      next
+        show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
+                    by (unfold tRAG_subtree_eq, auto simp:subtree_def)
+      next
+        show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
+                      (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
+        proof -
+          have "?L = the_preced (t @ s) `  threads (t @ s)" 
+                     by (unfold image_comp, rule image_cong, auto)
+          thus ?thesis using max_preced the_preced_def by auto
+        qed
+      qed
+      also have "... = ?R"
+        using th_cp_max th_cp_preced th_kept 
+              the_preced_def vat_t.max_cp_readys_threads by auto
+      finally show ?thesis .
+    qed 
+    -- {* Now, since @{term th'} holds the highest @{term cp} 
+          and we have already show it is in @{term readys},
+          it is @{term runing} by definition. *}
+    with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def) 
+  qed
+  -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
+  moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" 
+    using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
+  ultimately show ?thesis using that by metis
+qed
+
+text {*
+  Now it is easy to see there is always a thread to run by case analysis
+  on whether thread @{term th} is running: if the answer is Yes, the 
+  the running thread is obviously @{term th} itself; otherwise, the running
+  thread is the @{text th'} given by lemma @{thm th_blockedE}.
+*}
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)") 
+  case True thus ?thesis by auto
+next
+  case False
+  thus ?thesis using th_blockedE by auto
+qed
+
+end
+end
+
diff -r 0525670d8e6a -r 4763aa246dbd PIPBasics.thy
--- a/PIPBasics.thy	Fri Jan 29 09:46:07 2016 +0800
+++ b/PIPBasics.thy	Fri Jan 29 10:51:52 2016 +0800
@@ -1,7 +1,160 @@
-theory PIPBasics
-imports PIPDefs 
+theory CpsG
+imports PIPDefs
 begin
 
+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
+
+lemma Max_fg_mono:
+  assumes "finite A"
+  and "\<forall> a \<in> A. f a \<le> g a"
+  shows "Max (f ` A) \<le> Max (g ` A)"
+proof(cases "A = {}")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  show ?thesis
+  proof(rule Max.boundedI)
+    from assms show "finite (f ` A)" by auto
+  next
+    from False show "f ` A \<noteq> {}" by auto
+  next
+    fix fa
+    assume "fa \<in> f ` A"
+    then obtain a where h_fa: "a \<in> A" "fa = f a" by auto
+    show "fa \<le> Max (g ` A)"
+    proof(rule Max_ge_iff[THEN iffD2])
+      from assms show "finite (g ` A)" by auto
+    next
+      from False show "g ` A \<noteq> {}" by auto
+    next
+      from h_fa have "g a \<in> g ` A" by auto
+      moreover have "fa \<le> g a" using h_fa assms(2) by auto
+      ultimately show "\<exists>a\<in>g ` A. fa \<le> a" by auto
+    qed
+  qed
+qed 
+
+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
+
+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
+
+lemma birth_time_lt:  
+  assumes "s \<noteq> []"
+  shows "last_set th s < length s"
+  using assms
+proof(induct s)
+  case (Cons a s)
+  show ?case
+  proof(cases "s \<noteq> []")
+    case False
+    thus ?thesis
+      by (cases a, auto)
+  next
+    case True
+    show ?thesis using Cons(1)[OF True]
+      by (cases a, auto)
+  qed
+qed simp
+
+lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
+  by (induct s, auto)
+
+lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
+  by (drule_tac th_in_ne, unfold preced_def, auto intro: birth_time_lt)
+
+lemma eq_RAG: 
+  "RAG (wq s) = RAG s"
+  by (unfold cs_RAG_def s_RAG_def, auto)
+
+lemma waiting_holding:
+  assumes "waiting (s::state) th cs"
+  obtains th' where "holding s th' cs"
+proof -
+  from assms[unfolded s_waiting_def, folded wq_def]
+  obtain th' where "th' \<in> set (wq s cs)" "th' = hd (wq s cs)"
+    by (metis empty_iff hd_in_set list.set(1))
+  hence "holding s th' cs" 
+    by (unfold s_holding_def, fold wq_def, auto)
+  from that[OF this] show ?thesis .
+qed
+
+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 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
+
+(* ccc *)
+
+
 locale valid_trace = 
   fixes s
   assumes vt : "vt s"
@@ -16,6 +169,105 @@
 
 end
 
+locale valid_trace_create = valid_trace_e + 
+  fixes th prio
+  assumes is_create: "e = Create th prio"
+
+locale valid_trace_exit = valid_trace_e + 
+  fixes th
+  assumes is_exit: "e = Exit th"
+
+locale valid_trace_p = valid_trace_e + 
+  fixes th cs
+  assumes is_p: "e = P th cs"
+
+locale valid_trace_v = valid_trace_e + 
+  fixes th cs
+  assumes is_v: "e = V th cs"
+begin
+  definition "rest = tl (wq s cs)"
+  definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+end
+
+locale valid_trace_v_n = valid_trace_v +
+  assumes rest_nnl: "rest \<noteq> []"
+
+locale valid_trace_v_e = valid_trace_v +
+  assumes rest_nil: "rest = []"
+
+locale valid_trace_set= valid_trace_e + 
+  fixes th prio
+  assumes is_set: "e = Set th prio"
+
+context valid_trace
+begin
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes "PP []"
+     and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
+                   PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
+     shows "PP s"
+proof(induct rule:vt.induct[OF vt, case_names Init Step])
+  case Init
+  from assms(1) show ?case .
+next
+  case (Step s e)
+  show ?case
+  proof(rule assms(2))
+    show "valid_trace_e s e" using Step by (unfold_locales, auto)
+  next
+    show "PP s" using Step by simp
+  next
+    show "PIP s e" using Step by simp
+  qed
+qed
+
+lemma  vt_moment: "\<And> t. vt (moment t s)"
+proof(induct rule:ind)
+  case Nil
+  thus ?case by (simp add:vt_nil)
+next
+  case (Cons s e t)
+  show ?case
+  proof(cases "t \<ge> length (e#s)")
+    case True
+    from True have "moment t (e#s) = e#s" by simp
+    thus ?thesis using Cons
+      by (simp add:valid_trace_def valid_trace_e_def, auto)
+  next
+    case False
+    from Cons have "vt (moment t s)" by simp
+    moreover have "moment t (e#s) = moment t s"
+    proof -
+      from False have "t \<le> length s" by simp
+      from moment_app [OF this, of "[e]"] 
+      show ?thesis by simp
+    qed
+    ultimately show ?thesis by simp
+  qed
+qed
+
+lemma finite_threads:
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, auto)
+
+locale valid_moment = valid_trace + 
+  fixes i :: nat
+
+sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
+  by (unfold_locales, insert vt_moment, auto)
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+  by  (unfold s_waiting_def cs_waiting_def wq_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def wq_def cs_holding_def, simp)
+
 lemma runing_ready: 
   shows "runing s \<subseteq> readys s"
   unfolding runing_def readys_def
@@ -26,7 +278,7 @@
   unfolding readys_def
   by auto
 
-lemma wq_v_neq:
+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)
 
@@ -40,6 +292,210 @@
 context valid_trace
 begin
 
+lemma runing_wqE:
+  assumes "th \<in> runing s"
+  and "th \<in> set (wq s cs)"
+  obtains rest where "wq s cs = th#rest"
+proof -
+  from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
+    by (meson list.set_cases)
+  have "th' = th"
+  proof(rule ccontr)
+    assume "th' \<noteq> th"
+    hence "th \<noteq> hd (wq s cs)" using eq_wq by auto 
+    with assms(2)
+    have "waiting s th cs" 
+      by (unfold s_waiting_def, fold wq_def, auto)
+    with assms show False 
+      by (unfold runing_def readys_def, auto)
+  qed
+  with eq_wq that show ?thesis by metis
+qed
+
+end
+
+context valid_trace_create
+begin
+
+lemma wq_neq_simp [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_create wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace_exit
+begin
+
+lemma wq_neq_simp [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_exit wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace_p
+begin
+
+lemma wq_neq_simp [simp]:
+  assumes "cs' \<noteq> cs"
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_p wq_def
+  by (auto simp:Let_def)
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_p]
+  show ?thesis by (cases, simp)
+qed
+
+lemma 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
+  assume otherwise: "th \<in> set (wq s cs)"
+  from runing_wqE[OF runing_th_s this]
+  obtain rest where eq_wq: "wq s cs = th#rest" by blast
+  with otherwise
+  have "holding s th cs"
+    by (unfold s_holding_def, fold wq_def, simp)
+  hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
+    by (unfold s_RAG_def, fold holding_eq, auto)
+  from pip_e[unfolded is_p]
+  show False
+  proof(cases)
+    case (thread_P)
+    with cs_th_RAG show ?thesis by auto
+  qed
+qed
+
+lemma wq_es_cs: 
+  "wq (e#s) cs =  wq s cs @ [th]"
+  by (unfold is_p wq_def, auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+proof(cases "cs' = cs")
+  case True
+  show ?thesis using True assms th_not_in_wq
+    by (unfold True wq_es_cs, auto)
+qed (insert assms, simp)
+
+end
+
+context valid_trace_v
+begin
+
+lemma wq_neq_simp [simp]:
+  assumes "cs' \<noteq> cs"
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_v wq_def
+  by (auto simp:Let_def)
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis by (cases, simp)
+qed
+
+lemma th_not_waiting: 
+  "\<not> waiting s th c"
+proof -
+  have "th \<in> readys s"
+    using runing_ready runing_th_s by blast 
+  thus ?thesis
+    by (unfold readys_def, auto)
+qed
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+lemma wq_s_cs:
+  "wq s cs = th#rest"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis
+  proof(cases)
+    case (thread_V)
+    from this(2) show ?thesis
+      by (unfold rest_def s_holding_def, fold wq_def,
+                 metis empty_iff list.collapse list.set(1))
+  qed
+qed
+
+lemma wq_es_cs:
+  "wq (e#s) cs = wq'"
+ using wq_s_cs[unfolded wq_def]
+ by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) 
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+proof(cases "cs' = cs")
+  case True
+  show ?thesis
+  proof(unfold True wq_es_cs wq'_def, rule someI2)
+    show "distinct rest \<and> set rest = set rest"
+        using assms[unfolded True wq_s_cs] by auto
+  qed simp
+qed (insert assms, simp)
+
+end
+
+context valid_trace_set
+begin
+
+lemma wq_neq_simp [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_set wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace
+begin
+
 lemma actor_inv: 
   assumes "PIP s e"
   and "\<not> isCreate e"
@@ -47,94 +503,49 @@
   using assms
   by (induct, auto)
 
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
-  assumes "PP []"
-     and "(\<And>s e. valid_trace s \<Longrightarrow> valid_trace (e#s) \<Longrightarrow>
-                   PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
-     shows "PP s"
-proof(rule vt.induct[OF vt])
-  from assms(1) show "PP []" .
-next
-  fix s e
-  assume h: "vt s" "PP s" "PIP s e"
-  show "PP (e # s)"
-  proof(cases rule:assms(2))
-    from h(1) show v1: "valid_trace s" by (unfold_locales, simp)
-  next
-    from h(1,3) have "vt (e#s)" by auto
-    thus "valid_trace (e # s)" by (unfold_locales, simp)
-  qed (insert h, auto)
-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) 
 
 lemma wq_distinct: "distinct (wq s cs)"
 proof(induct rule:ind)
   case (Cons s e)
-  from Cons(4,3)
+  interpret vt_e: valid_trace_e s e using Cons by simp
   show ?case 
-  proof(induct)
-    case (thread_P th s cs1)
-    show ?case 
-    proof(cases "cs = cs1")
-      case True
-      thus ?thesis (is "distinct ?L")
-      proof - 
-        have "?L = wq_fun (schs s) cs1 @ [th]" using True
-          by (simp add:wq_def wf_def Let_def split:list.splits)
-        moreover have "distinct ..."
-        proof -
-          have "th \<notin> set (wq_fun (schs s) cs1)"
-          proof
-            assume otherwise: "th \<in> set (wq_fun (schs s) cs1)"
-            from runing_head[OF thread_P(1) this]
-            have "th = hd (wq_fun (schs s) cs1)" .
-            hence "(Cs cs1, Th th) \<in> (RAG s)" using otherwise
-              by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
-            with thread_P(2) show False by auto
-          qed
-          moreover have "distinct (wq_fun (schs s) cs1)"
-              using True thread_P wq_def by auto 
-          ultimately show ?thesis by auto
-        qed
-        ultimately show ?thesis by simp
-      qed
-    next
-      case False
-      with thread_P(3)
-      show ?thesis
-        by (auto simp:wq_def wf_def Let_def split:list.splits)
-    qed
+  proof(cases e)
+    case (Create th prio)
+    interpret vt_create: valid_trace_create s e th prio 
+      using Create by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept) 
+  next
+    case (Exit th)
+    interpret vt_exit: valid_trace_exit s e th  
+        using Exit by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept) 
   next
-    case (thread_V th s cs1)
-    thus ?case
-    proof(cases "cs = cs1")
-      case True
-      show ?thesis (is "distinct ?L")
-      proof(cases "(wq s cs)")
-        case Nil
-        thus ?thesis
-          by (auto simp:wq_def wf_def Let_def split:list.splits)
-      next
-        case (Cons w_hd w_tl)
-        moreover have "distinct (SOME q. distinct q \<and> set q = set w_tl)"
-        proof(rule someI2)
-          from thread_V(3)[unfolded Cons]
-          show  "distinct w_tl \<and> set w_tl = set w_tl" by auto
-        qed auto
-        ultimately show ?thesis
-          by (auto simp:wq_def wf_def Let_def True split:list.splits)
-      qed 
-    next
-      case False
-      with thread_V(3)
-      show ?thesis
-        by (auto simp:wq_def wf_def Let_def split:list.splits)
-    qed
-  qed (insert Cons, auto simp: wq_def Let_def split:list.splits)
+    case (P th cs)
+    interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) 
+  next
+    case (V th cs)
+    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) 
+  next
+    case (Set th prio)
+    interpret vt_set: valid_trace_set s e th prio
+        using Set by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) 
+  qed
 qed (unfold wq_def Let_def, simp)
 
 end
 
-
 context valid_trace_e
 begin
 
@@ -145,7 +556,7 @@
   This is a kind of confirmation that our modelling is correct.
 *}
 
-lemma block_pre: 
+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"
@@ -175,117 +586,44 @@
   thus ?thesis by auto
 qed (insert assms, auto simp:wq_def Let_def split:if_splits)
 
-end
-
-text {*
-  The following lemmas is also obvious and shallow. It says
-  that only running thread can request for a critical resource 
-  and that the requested resource must be one which is
-  not current held by the thread.
-*}
-
-lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow> 
-  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (RAG s)^+"
-apply (ind_cases "vt ((P thread cs)#s)")
-apply (ind_cases "step s (P thread cs)")
-by auto
-
-lemma abs1:
-  assumes ein: "e \<in> set es"
-  and neq: "hd es \<noteq> hd (es @ [x])"
-  shows "False"
-proof -
-  from ein have "es \<noteq> []" by auto
-  then obtain e ess where "es = e # ess" by (cases es, auto)
-  with neq show ?thesis by auto
-qed
-
-lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
-  by (cases es, auto)
-
-inductive_cases evt_cons: "vt (a#s)"
-
-context valid_trace_e
-begin
-
-lemma abs2:
-  assumes inq: "thread \<in> set (wq s cs)"
-  and nh: "thread = hd (wq s cs)"
-  and qt: "thread \<noteq> hd (wq (e#s) cs)"
-  and inq': "thread \<in> set (wq (e#s) cs)"
-  shows "False"
-proof -
-  from vt_e assms show "False"
-    apply (cases e)
-    apply ((simp split:if_splits add:Let_def wq_def)[1])+
-    apply (insert abs1, fast)[1]
-    apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)
-  proof -
-    fix th qs
-    assume vt: "vt (V th cs # s)"
-      and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
-      and eq_wq: "wq_fun (schs s) cs = thread # qs"
-    show "False"
-    proof -
-      from wq_distinct[of cs]
-        and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
-      moreover have "thread \<in> set qs"
-      proof -
-        have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
-        proof(rule someI2)
-          from wq_distinct [of cs]
-          and eq_wq [folded wq_def]
-          show "distinct qs \<and> set qs = set qs" by auto
-        next
-          fix x assume "distinct x \<and> set x = set qs"
-          thus "set x = set qs" by auto
-        qed
-        with th_in show ?thesis by auto
-      qed
-      ultimately show ?thesis by auto
+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
-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
-lemma  vt_moment: "\<And> t. vt (moment t s)"
-proof(induct rule:ind)
-  case Nil
-  thus ?case by (simp add:vt_nil)
-next
-  case (Cons s e t)
-  show ?case
-  proof(cases "t \<ge> length (e#s)")
-    case True
-    from True have "moment t (e#s) = e#s" by simp
-    thus ?thesis using Cons
-      by (simp add:valid_trace_def)
-  next
-    case False
-    from Cons have "vt (moment t s)" by simp
-    moreover have "moment t (e#s) = moment t s"
-    proof -
-      from False have "t \<le> length s" by simp
-      from moment_app [OF this, of "[e]"] 
-      show ?thesis by simp
-    qed
-    ultimately show ?thesis by simp
-  qed
-qed
-end
-
-locale valid_moment = valid_trace + 
-  fixes i :: nat
-
-sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
-  by (unfold_locales, insert vt_moment, auto)
-
-context valid_trace
-begin
 
 
 text {* (* ddd *)
@@ -321,7 +659,7 @@
   make any request and get blocked the second time: Contradiction.
 *}
 
-lemma waiting_unique_pre: (* ccc *)
+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)"
@@ -329,35 +667,101 @@
   and neq12: "cs1 \<noteq> cs2"
   shows "False"
 proof -
-  let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  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>(thread \<in> set (wq (moment t1 s) cs1) \<and>
-        thread \<noteq> hd (wq (moment t1 s) cs1))"
-    and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
-             thread \<noteq> hd (wq (moment i' s) cs1))" by auto
+    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>(thread \<in> set (wq (moment t2 s) cs2) \<and>
-        thread \<noteq> hd (wq (moment t2 s) cs2))"
-    and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
-             thread \<noteq> hd (wq (moment i' s) cs2))" by auto
+    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 lt12: "t1 < t2"
+    { 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 "t2 < ?t3" by simp
+      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
+           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 
@@ -365,119 +769,38 @@
         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
+          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
-        from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
-          by auto 
-        from vt_e.abs2 [OF True eq_th h2 h1]
-        show ?thesis by auto
-      next
-        case False
-        from vt_e.block_pre[OF False h1]
-        have "e = P thread cs2" .
-        with vt_e.vt_e have "vt ((P thread cs2)# moment t2 s)" by simp
-        from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
-        with runing_ready have "thread \<in> readys (moment t2 s)" by auto
-        with nn1 [rule_format, OF lt12]
-        show ?thesis  by (simp add:readys_def wq_def s_waiting_def, auto)
-      qed
-    } moreover {
-      assume lt12: "t2 < t1"
-      let ?t3 = "Suc t1"
-      from lt1 have le_t3: "?t3 \<le> length s" by auto
-      from moment_plus [OF this] 
-      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
-      have lt_t3: "t1 < ?t3" by simp
-      from nn1 [rule_format, OF this] and eq_m
-      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
-        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
-      have "vt  (e#moment t1 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 t1 s" e
-        by (unfold_locales, auto, cases, auto)
-      have ?thesis
-      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
-        case True
-        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
-          by auto
-        from vt_e.abs2 True eq_th h2 h1
-        show ?thesis by auto
+        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
-        from vt_e.block_pre [OF False h1]
-        have "e = P thread cs1" .
-        with vt_e.vt_e have "vt ((P thread cs1)# moment t1 s)" by simp
-        from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
-        with runing_ready have "thread \<in> readys (moment t1 s)" by auto
-        with nn2 [rule_format, OF lt12]
-        show ?thesis  by (simp add:readys_def wq_def s_waiting_def, auto)
+        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+        thus ?thesis by auto
       qed
-    } moreover {
-      assume eqt12: "t1 = t2"
-      let ?t3 = "Suc t1"
-      from lt1 have le_t3: "?t3 \<le> length s" by auto
-      from moment_plus [OF this] 
-      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
-      have lt_t3: "t1 < ?t3" by simp
-      from nn1 [rule_format, OF this] and eq_m
-      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
-        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
-      have vt_e: "vt (e#moment t1 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 t1 s" e
-        by (unfold_locales, auto, cases, auto)
-      have ?thesis
+      moreover have "e = V thread cs1 \<or> e = P thread cs1"
       proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
         case True
-        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
-          by auto
-        from vt_e.abs2 [OF True eq_th h2 h1]
-        show ?thesis by auto
+        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
-        from vt_e.block_pre [OF False h1]
-        have eq_e1: "e = P thread cs1" .
-        have lt_t3: "t1 < ?t3" by simp
-        with eqt12 have "t2 < ?t3" by simp
-        from nn2 [rule_format, OF this] and eq_m and eqt12
-        have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-          h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-        show ?thesis
-        proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-          case True
-          from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
-            by auto
-          from vt_e and eqt12 have "vt (e#moment t2 s)" by simp 
-          then interpret vt_e2: valid_trace_e "moment t2 s" e
-            by (unfold_locales, auto, cases, auto)
-          from vt_e2.abs2 [OF True eq_th h2 h1]
-          show ?thesis .
-        next
-          case False
-          have "vt (e#moment t2 s)"
-          proof -
-            from vt_moment eqt12
-            have "vt (moment (Suc t2) s)" by auto
-            with eq_m eqt12 show ?thesis by simp
-          qed
-          then interpret vt_e2: valid_trace_e "moment t2 s" e
-            by (unfold_locales, auto, cases, auto)
-          from vt_e2.block_pre [OF False h1]
-          have "e = P thread cs2" .
-          with eq_e1 neq12 show ?thesis by auto
-        qed
+        have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
+        thus ?thesis by auto
       qed
-    } ultimately show ?thesis by arith
+      ultimately have ?thesis using neq12 by auto
+    } ultimately show ?thesis using nat_neq_iff by blast 
   qed
 qed
 
@@ -489,9 +812,9 @@
   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
+  using waiting_unique_pre assms
+  unfolding wq_def s_waiting_def
+  by auto
 
 end
 
@@ -507,7 +830,6 @@
   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)
@@ -528,7 +850,7 @@
   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"
@@ -540,98 +862,6 @@
   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"}
@@ -642,598 +872,1404 @@
 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)
 
-
-text {* 
-  The following lemmas are used in the proof of 
-  lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed
-  by @{text "V"}-events. 
-  However, since our model is very concise, such  seemingly obvious lemmas need to be derived from scratch,
-  starting from the model definitions.
-*}
-lemma step_v_hold_inv[elim_format]:
-  "\<And>c t. \<lbrakk>vt (V th cs # s); 
-          \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> 
-            next_th s th cs t \<and> c = cs"
+lemma (in valid_trace_exit)
+   RAG_unchanged: "(RAG (e # s)) = RAG s"
+   by (unfold is_exit RAG_exit_unchanged, simp)
+
+context valid_trace_v
+begin
+
+lemma distinct_rest: "distinct rest"
+  by (simp add: distinct_tl rest_def wq_distinct)
+
+lemma holding_cs_eq_th:
+  assumes "holding s t cs"
+  shows "t = th"
 proof -
-  fix c t
-  assume vt: "vt (V th cs # s)"
-    and nhd: "\<not> holding (wq s) t c"
-    and hd: "holding (wq (V th cs # s)) t c"
-  show "next_th s th cs t \<and> c = cs"
-  proof(cases "c = cs")
-    case False
-    with nhd hd show ?thesis
-      by (unfold cs_holding_def wq_def, auto simp:Let_def)
-  next
-    case True
-    with step_back_step [OF vt] 
-    have "step s (V th c)" by simp
-    hence "next_th s th cs t"
-    proof(cases)
-      assume "holding s th c"
-      with nhd hd show ?thesis
-        apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
-               auto simp:Let_def split:list.splits if_splits)
-        proof -
-          assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
-          moreover have "\<dots> = set []"
-          proof(rule someI2)
-            show "distinct [] \<and> [] = []" by auto
-          next
-            fix x assume "distinct x \<and> x = []"
-            thus "set x = set []" by auto
-          qed
-          ultimately show False by auto
-        next
-          assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
-          moreover have "\<dots> = set []"
-          proof(rule someI2)
-            show "distinct [] \<and> [] = []" by auto
-          next
-            fix x assume "distinct x \<and> x = []"
-            thus "set x = set []" by auto
-          qed
-          ultimately show False by auto
-        qed
-    qed
-    with True show ?thesis by auto
+  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
 
-text {* 
-  The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be
-  derived from scratch, which confirms the correctness of the definition of @{text "next_th"}.
-*}
-lemma step_v_wait_inv[elim_format]:
-    "\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
-           \<rbrakk>
-          \<Longrightarrow> (next_th s th cs t \<and> cs = c)"
-proof -
-  fix t c 
-  assume vt: "vt (V th cs # s)"
-    and nw: "\<not> waiting (wq (V th cs # s)) t c"
-    and wt: "waiting (wq s) t c"
-  from vt interpret vt_v: valid_trace_e s "V th cs" 
-    by  (cases, unfold_locales, simp)
-  show "next_th s th cs t \<and> cs = c"
-  proof(cases "cs = c")
-    case False
-    with nw wt show ?thesis
-      by (auto simp:cs_waiting_def wq_def Let_def)
-  next
+lemma distinct_wq': "distinct wq'"
+  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)
+    
+lemma th'_in_inv:
+  assumes "th' \<in> set wq'"
+  shows "th' \<in> set rest"
+  using assms set_wq' by simp
+
+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
-    from nw[folded True] wt[folded True]
-    have "next_th s th cs t"
-      apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
-    proof -
-      fix a list
-      assume t_in: "t \<in> set list"
-        and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
-        and eq_wq: "wq_fun (schs s) cs = a # list"
-      have " set (SOME q. distinct q \<and> set q = set list) = set list"
-      proof(rule someI2)
-        from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
-        show "distinct list \<and> set list = set list" by auto
-      next
-        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
-          by auto
-      qed
-      with t_ni and t_in show "a = th" by auto
-    next
-      fix a list
-      assume t_in: "t \<in> set list"
-        and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
-        and eq_wq: "wq_fun (schs s) cs = a # list"
-      have " set (SOME q. distinct q \<and> set q = set list) = set list"
-      proof(rule someI2)
-        from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
-        show "distinct list \<and> set list = set list" by auto
-      next
-        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
-          by auto
-      qed
-      with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto
-    next
-      fix a list
-      assume eq_wq: "wq_fun (schs s) cs = a # list"
-      from step_back_step[OF vt]
-      show "a = th"
-      proof(cases)
-        assume "holding s th cs"
-        with eq_wq show ?thesis
-          by (unfold s_holding_def wq_def, auto)
-      qed
-    qed
-    with True show ?thesis by simp
+    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 step_v_not_wait[consumes 3]:
-  "\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
-  by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
-
-lemma step_v_release:
-  "\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
+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 -
-  assume vt: "vt (V th cs # s)"
-    and hd: "holding (wq (V th cs # s)) th cs"
-  from vt interpret vt_v: valid_trace_e s "V th cs"
-    by (cases, unfold_locales, simp+)
-  from step_back_step [OF vt] and hd
-  show "False"
-  proof(cases)
-    assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"
-    thus ?thesis
-      apply (unfold s_holding_def wq_def cs_holding_def)
-      apply (auto simp:Let_def split:list.splits)
-    proof -
-      fix list
-      assume eq_wq[folded wq_def]: 
-        "wq_fun (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
-      and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
-            \<in> set (SOME q. distinct q \<and> set q = set list)"
-      have "set (SOME q. distinct q \<and> set q = set list) = set list"
-      proof(rule someI2)
-        from vt_v.wq_distinct[of cs] and eq_wq
-        show "distinct list \<and> set list = set list" by auto
+  have "t \<in> set wq'" 
+  proof(unfold wq'_def, rule someI2)
+    show "distinct rest \<and> set rest = set rest"
+          by (simp add: distinct_rest)
+  next
+    fix x
+    assume "distinct x \<and> set x = set rest"
+    moreover have "t \<in> set rest"
+        using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto 
+    ultimately show "t \<in> set x" by simp
+  qed
+  moreover have "t \<noteq> hd wq'"
+    using assms(2) taker_def by auto 
+  ultimately show ?thesis
+    by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+     |    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
+  have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
+  hence "t \<noteq> taker" by (simp add: taker_def) 
+  moreover hence "t \<noteq> th" using assms neq_t_th by blast 
+  moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv) 
+  ultimately have "waiting s t cs"
+    by (metis cs_waiting_def list.distinct(2) list.sel(1) 
+                list.set_sel(2) rest_def waiting_eq wq_s_cs)  
+  show ?thesis using that(2)
+  using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto   
+qed
+
+lemma holding_esI1:
+  assumes "c = cs"
+  and "t = taker"
+  shows "holding (e#s) t c"
+  by (unfold assms, simp add: holding_taker)
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c = cs" "t = taker"
+      | "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from assms[unfolded True, unfolded s_holding_def, 
+             folded wq_def, unfolded wq_es_cs]
+  have "t = taker" by (simp add: taker_def) 
+  from that(1)[OF True this] show ?thesis .
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that(2)[OF False this] show ?thesis .
+qed
+
+end 
+
+
+context valid_trace_v_e
+begin
+
+lemma nil_wq': "wq' = []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x = []" using rest_nil by auto
+qed 
+
+lemma no_taker: 
+  assumes "next_th s th cs taker"
+  shows "False"
+proof -
+  from assms[unfolded next_th_def]
+  obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
+    by auto
+  thus ?thesis using rest_def rest_nil by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
+  using no_taker by auto
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
+  using no_taker by auto
+   
+lemma no_holding:
+  assumes "holding (e#s) taker cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_holding_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma no_waiting:
+  assumes "waiting (e#s) t cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma waiting_esI2:
+  assumes "waiting s t c"
+  shows "waiting (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms
+    using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto 
+  from waiting_esI1[OF assms this]
+  show ?thesis .
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from no_waiting[OF assms[unfolded True]]
+  show ?thesis by auto
+qed
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from no_holding[OF assms[unfolded True]] 
+  show ?thesis by auto
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that[OF False this] show ?thesis .
+qed
+
+end 
+
+lemma rel_eqI:
+  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
+  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
+  shows "A = B"
+  using assms by auto
+
+lemma in_RAG_E:
+  assumes "(n1, n2) \<in> RAG (s::state)"
+  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
+      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
+  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
+  by auto
+  
+context valid_trace_v
+begin
+
+lemma RAG_es:
+  "RAG (e # s) =
+   RAG s - {(Cs cs, Th th)} -
+     {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+     {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R"
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_n.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
       next
-        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
-          by auto
+        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
-      moreover have "distinct  (hd (SOME q. distinct q \<and> set q = set list) # list)"
-      proof -
-        from vt_v.wq_distinct[of cs] and eq_wq
-        show ?thesis by auto
+    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
-      moreover note eq_wq and hd_in
-      ultimately show "False" by auto
+    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
-qed
-
-lemma step_v_get_hold:
-  "\<And>th'. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
-  apply (unfold cs_holding_def next_th_def wq_def,
-         auto simp:Let_def)
-proof -
-  fix rest
-  assume vt: "vt (V th cs # s)"
-    and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest"
-    and nrest: "rest \<noteq> []"
-    and ni: "hd (SOME q. distinct q \<and> set q = set rest)
-            \<notin> set (SOME q. distinct q \<and> set q = set rest)"
-  from vt interpret vt_v: valid_trace_e s "V th cs"
-    by (cases, unfold_locales, simp+)
-  have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-  proof(rule someI2)
-    from vt_v.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"
-    hence "set x = set rest" by auto
-    with nrest
-    show "x \<noteq> []" by (case_tac x, auto)
-  qed
-  with ni show "False" by auto
-qed
-
-lemma step_v_release_inv[elim_format]:
-"\<And>c t. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> 
-  c = cs \<and> t = th"
-  apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
-  proof -
-    fix a list
-    assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
-    from step_back_step [OF vt] show "a = th"
-    proof(cases)
-      assume "holding s th cs" with eq_wq
-      show ?thesis
-        by (unfold s_holding_def wq_def, auto)
-    qed
-  next
-    fix a list
-    assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
-    from step_back_step [OF vt] show "a = th"
-    proof(cases)
-      assume "holding s th cs" with eq_wq
-      show ?thesis
-        by (unfold s_holding_def wq_def, auto)
+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
-
-lemma step_v_waiting_mono:
-  "\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
-proof -
-  fix t c
-  let ?s' = "(V th cs # s)"
-  assume vt: "vt ?s'" 
-    and wt: "waiting (wq ?s') t c"
-  from vt interpret vt_v: valid_trace_e s "V th cs"
-    by (cases, unfold_locales, simp+)
-  show "waiting (wq s) t c"
-  proof(cases "c = cs")
-    case False
-    assume neq_cs: "c \<noteq> cs"
-    hence "waiting (wq ?s') t c = waiting (wq s) t c"
-      by (unfold cs_waiting_def wq_def, auto simp:Let_def)
-    with wt show ?thesis by simp
-  next
-    case True
-    with wt show ?thesis
-      apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
-    proof -
-      fix a list
-      assume not_in: "t \<notin> set list"
-        and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
-        and eq_wq: "wq_fun (schs s) cs = a # list"
-      have "set (SOME q. distinct q \<and> set q = set list) = set list"
-      proof(rule someI2)
-        from vt_v.wq_distinct [of cs]
-        and eq_wq[folded wq_def]
-        show "distinct list \<and> set list = set list" by auto
-      next
-        fix x assume "distinct x \<and> set x = set list"
-        thus "set x = set list" by auto
-      qed
-      with not_in is_in show "t = a" by auto
+   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
-      fix list
-      assume is_waiting: "waiting (wq (V th cs # s)) t cs"
-      and eq_wq: "wq_fun (schs s) cs = t # list"
-      hence "t \<in> set list"
-        apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
-      proof -
-        assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
-        moreover have "\<dots> = set list" 
-        proof(rule someI2)
-          from vt_v.wq_distinct [of cs]
-            and eq_wq[folded wq_def]
-          show "distinct list \<and> set list = set list" by auto
-        next
-          fix x assume "distinct x \<and> set x = set list" 
-          thus "set x = set list" by auto
-        qed
-        ultimately show "t \<in> set list" by simp
-      qed
-      with eq_wq and vt_v.wq_distinct [of cs, unfolded wq_def]
-      show False by auto
+      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
 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:
+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'}"
-  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)
+  {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof -
+  interpret vt_v: valid_trace_v s "V th cs"
+    using assms step_back_vt by (unfold_locales, auto) 
+  show ?thesis using vt_v.RAG_es .
+qed
+
+lemma (in valid_trace_create)
+  th_not_in_threads: "th \<notin> threads s"
+proof -
+  from pip_e[unfolded is_create]
+  show ?thesis by (cases, simp)
+qed
+
+lemma (in valid_trace_create)
+  threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
+  by (unfold is_create, simp)
+
+lemma (in valid_trace_exit)
+  threads_es [simp]: "threads (e#s) = threads s - {th}"
+  by (unfold is_exit, simp)
+
+lemma (in valid_trace_p)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_p, simp)
+
+lemma (in valid_trace_v)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_v, simp)
+
+lemma (in valid_trace_v)
+  th_not_in_rest[simp]: "th \<notin> set rest"
+proof
+  assume otherwise: "th \<in> set rest"
+  have "distinct (wq s cs)" by (simp add: wq_distinct)
+  from this[unfolded wq_s_cs] and otherwise
+  show False by auto
+qed
+
+lemma (in valid_trace_v)
+  set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
+proof(unfold wq_es_cs wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest)
+next
+  fix x
+  assume "distinct x \<and> set x = set rest"
+  thus "set x = set (wq s cs) - {th}" 
+      by (unfold wq_s_cs, simp)
+qed
+
+lemma (in valid_trace_exit)
+  th_not_in_wq: "th \<notin> set (wq s cs)"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis
+  by (cases, unfold holdents_def s_holding_def, fold wq_def, 
+             auto elim!:runing_wqE)
+qed
+
+lemma (in valid_trace) wq_threads: 
+  assumes "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+  using assms
+proof(induct rule:ind)
+  case (Nil)
+  thus ?case by (auto simp:wq_def)
+next
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case
+  proof(cases e)
+    case (Create th' prio')
+    interpret vt: valid_trace_create s e th' prio'
+      using Create by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems by auto
+  next
+    case (Exit th')
+    interpret vt: valid_trace_exit s e th'
+      using Exit by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto 
+  next
+    case (P th' cs')
+    interpret vt: valid_trace_p s e th' cs'
+      using P by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems readys_threads 
+        runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv 
+        by fastforce 
+  next
+    case (V th' cs')
+    interpret vt: valid_trace_v s e th' cs'
+      using V by (unfold_locales, simp)
+    show ?thesis using Cons
+      using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
+  next
+    case (Set th' prio)
+    interpret vt: valid_trace_set s e th' prio
+      using Set by (unfold_locales, simp)
+    show ?thesis using Cons.hyps(2) Cons.prems vt.is_set 
+        by (auto simp:wq_def Let_def)
+  qed
+qed 
 
 context valid_trace
 begin
 
-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)
+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'"
+  using assms
+  by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2) 
+      rotate1.simps(2) self_append_conv2 set_rotate1 
+        th_not_in_wq waiting_eq wq_es_cs wq_neq_simp)
+  
+lemma holding_kept:
+  assumes "holding s th' cs'"
+  shows "holding (e#s) th' cs'"
+proof(cases "cs' = cs")
+  case False
+  hence "wq (e#s) cs' = wq s cs'" by simp
+  with assms show ?thesis using cs_holding_def holding_eq by auto 
+next
+  case True
+  from assms[unfolded s_holding_def, folded wq_def]
+  obtain rest where eq_wq: "wq s cs' = th'#rest"
+    by (metis empty_iff list.collapse list.set(1)) 
+  hence "wq (e#s) cs' = th'#(rest@[th])"
+    by (simp add: True wq_es_cs) 
+  thus ?thesis
+    by (simp add: cs_holding_def holding_eq) 
+qed
+
+end
+
+locale valid_trace_p_h = valid_trace_p +
+  assumes we: "wq s cs = []"
+
+locale valid_trace_p_w = valid_trace_p +
+  assumes wne: "wq s cs \<noteq> []"
+begin
+
+definition "holder = hd (wq s cs)"
+definition "waiters = tl (wq s cs)"
+definition "waiters' = waiters @ [th]"
+
+lemma wq_s_cs: "wq s cs = holder#waiters"
+    by (simp add: holder_def waiters_def wne)
+    
+lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
+  by (simp add: wq_es_cs wq_s_cs)
+
+lemma waiting_es_th_cs: "waiting (e#s) th cs"
+  using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
+
+lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
+   by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
+
+lemma holding_esE:
+  assumes "holding (e#s) th' cs'"
+  obtains "holding s th' cs'"
+  using assms 
+proof(cases "cs' = cs")
+  case False
+  hence "wq (e#s) cs' = wq s cs'" by simp
+  with assms show ?thesis
+    using cs_holding_def holding_eq that by auto 
+next
+  case True
+  with assms show ?thesis
+  by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that 
+        wq_es_cs' wq_s_cs) 
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) th' cs'"
+  obtains "th' \<noteq> th" "waiting s th' cs'"
+     |  "th' = th" "cs' = cs"
+proof(cases "waiting s th' cs'")
+  case True
+  have "th' \<noteq> th"
+  proof
+    assume otherwise: "th' = th"
+    from True[unfolded this]
+    show False by (simp add: th_not_waiting) 
+  qed
+  from that(1)[OF this True] show ?thesis .
+next
+  case False
+  hence "th' = th \<and> cs' = cs"
+      by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) 
+        set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
+  with that(2) show ?thesis by metis
+qed
+
+lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R" 
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:waiting_esE)
+      case 1
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case 2
+      thus ?thesis using waiting(1,2) by auto
+    qed
+  next
+    case (holding th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:holding_esE)
+      case 1
+      with holding(1,2)
+      show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  qed
+next
+  fix n1 n2
+  assume "(n1, n2) \<in> ?R"
+  hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
+  thus "(n1, n2) \<in> ?L"
+  proof
+    assume "(n1, n2) \<in> RAG s"
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from waiting_kept[OF this(3)]
+      show ?thesis using waiting(1,2)
+         by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from holding_kept[OF this(3)]
+      show ?thesis using holding(1,2)
+         by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  next
+    assume "n1 = Th th \<and> n2 = Cs cs"
+    thus ?thesis using RAG_edge by auto
+  qed
+qed
+
+end
+
+context valid_trace_p_h
+begin
+
+lemma wq_es_cs': "wq (e#s) cs = [th]"
+  using wq_es_cs[unfolded we] by simp
+
+lemma holding_es_th_cs: 
+  shows "holding (e#s) th cs"
+proof -
+  from wq_es_cs'
+  have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto
+  thus ?thesis using cs_holding_def holding_eq by blast 
+qed
+
+lemma RAG_edge: "(Cs cs, Th th) \<in> RAG (e#s)"
+  by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto)
+
+lemma waiting_esE:
+  assumes "waiting (e#s) th' cs'"
+  obtains "waiting s th' cs'"
+  using assms
+  by (metis cs_waiting_def event.distinct(15) is_p list.sel(1) 
+        set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv)
+  
+lemma holding_esE:
+  assumes "holding (e#s) th' cs'"
+  obtains "cs' \<noteq> cs" "holding s th' cs'"
+    | "cs' = cs" "th' = th"
+proof(cases "cs' = cs")
+  case True
+  from held_unique[OF holding_es_th_cs assms[unfolded True]]
+  have "th' = th" by simp
+  from that(2)[OF True this] show ?thesis .
+next
+  case False
+  have "holding s th' cs'" using assms
+    using False cs_holding_def holding_eq by auto
+  from that(1)[OF False this] show ?thesis .
+qed
+
+lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Cs cs, Th th)}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R" 
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:waiting_esE)
+      case 1
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    qed
+  next
+    case (holding th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:holding_esE)
+      case 1
+      with holding(1,2)
+      show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+    next
+      case 2
+      with holding(1,2) show ?thesis by auto
+    qed
+  qed
+next
+  fix n1 n2
+  assume "(n1, n2) \<in> ?R"
+  hence "(n1, n2) \<in> RAG s \<or> (n1 = Cs cs \<and> n2 = Th th)" by auto
+  thus "(n1, n2) \<in> ?L"
+  proof
+    assume "(n1, n2) \<in> RAG s"
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from waiting_kept[OF this(3)]
+      show ?thesis using waiting(1,2)
+         by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from holding_kept[OF this(3)]
+      show ?thesis using holding(1,2)
+         by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  next
+    assume "n1 = Cs cs \<and> n2 = Th th"
+    with holding_es_th_cs
+    show ?thesis 
+      by (unfold s_RAG_def, fold holding_eq, auto)
+  qed
+qed
+
+end
+
+context valid_trace_p
+begin
+
+lemma RAG_es': "RAG (e # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+                                                  else RAG s \<union> {(Th th, Cs cs)})"
+proof(cases "wq s cs = []")
+  case True
+  interpret vt_p: valid_trace_p_h using True
     by (unfold_locales, simp)
-  assume ih: "acyclic (RAG s)"
-    and stp: "step s e"
-    and vt: "vt s"
+  show ?thesis by (simp add: vt_p.RAG_es vt_p.we) 
+next
+  case False
+  interpret vt_p: valid_trace_p_w using False
+    by (unfold_locales, simp)
+  show ?thesis by (simp add: vt_p.RAG_es vt_p.wne) 
+qed
+
+end
+
+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)
+
+lemma (in valid_trace_v_n) finite_holding_set:
+  "finite {(Cs cs, Th th') |th'. next_th s th cs th'}"
+    by (simp add: holding_set_eq)
+
+lemma (in valid_trace_v_e) finite_waiting_set:
+  "finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
+    by (simp add: waiting_set_eq)
+
+lemma (in valid_trace_v_e) finite_holding_set:
+  "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 
+
+lemma 
+  acylic_RAG_kept:
+  assumes "acyclic (RAG s)"
+  shows "acyclic (RAG (e#s))"
+proof(rule acyclic_subset[OF assms])
+  show "RAG (e # s) \<subseteq> RAG s"
+      by (unfold RAG_es waiting_set_eq holding_set_eq, auto)
+qed
+
+end
+
+context valid_trace_v_n
+begin 
+
+lemma waiting_taker: "waiting s taker cs"
+  apply (unfold s_waiting_def, fold wq_def, unfold wq_s_cs taker_def)
+  using eq_wq' th'_in_inv wq'_def by fastforce
+
+lemma 
+  acylic_RAG_kept:
+  assumes "acyclic (RAG s)"
+  shows "acyclic (RAG (e#s))"
+proof -
+  have "acyclic ((RAG s - {(Cs cs, Th th)} - {(Th taker, Cs cs)}) \<union> 
+                 {(Cs cs, Th taker)})" (is "acyclic (?A \<union> _)")
+  proof -
+    from assms
+    have "acyclic ?A"
+       by (rule acyclic_subset, auto)
+    moreover have "(Th taker, Cs cs) \<notin> ?A^*"
+    proof
+      assume otherwise: "(Th taker, Cs cs) \<in> ?A^*"
+      hence "(Th taker, Cs cs) \<in> ?A^+"
+        by (unfold rtrancl_eq_or_trancl, auto)
+      from tranclD[OF this]
+      obtain cs' where h: "(Th taker, Cs cs') \<in> ?A" 
+                          "(Th taker, Cs cs') \<in> RAG s"
+        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]
+      have "cs' = cs" .
+      from h(1)[unfolded this] show False by auto
+    qed
+    ultimately show ?thesis by auto
+  qed
+  thus ?thesis 
+    by (unfold RAG_es waiting_set_eq holding_set_eq, simp)
+qed
+
+end
+
+context valid_trace_p_h
+begin
+
+lemma 
+  acylic_RAG_kept:
+  assumes "acyclic (RAG s)"
+  shows "acyclic (RAG (e#s))"
+proof -
+  have "acyclic (RAG s \<union> {(Cs cs, Th th)})" (is "acyclic (?A \<union> _)") 
+  proof -
+    from assms
+    have "acyclic ?A"
+       by (rule acyclic_subset, auto)
+    moreover have "(Th th, Cs cs) \<notin> ?A^*"
+    proof
+      assume otherwise: "(Th th, Cs cs) \<in> ?A^*"
+      hence "(Th th, Cs cs) \<in> ?A^+"
+        by (unfold rtrancl_eq_or_trancl, auto)
+      from tranclD[OF this]
+      obtain cs' where h: "(Th th, Cs cs') \<in> RAG s"
+        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
+    qed
+    ultimately show ?thesis by auto
+  qed
+  thus ?thesis by (unfold RAG_es, simp)
+qed
+
+end
+
+context valid_trace_p_w
+begin
+
+lemma 
+  acylic_RAG_kept:
+  assumes "acyclic (RAG s)"
+  shows "acyclic (RAG (e#s))"
+proof -
+  have "acyclic (RAG s \<union> {(Th th, Cs cs)})" (is "acyclic (?A \<union> _)") 
+  proof -
+    from assms
+    have "acyclic ?A"
+       by (rule acyclic_subset, auto)
+    moreover have "(Cs cs, Th th) \<notin> ?A^*"
+    proof
+      assume otherwise: "(Cs cs, Th th) \<in> ?A^*"
+      from pip_e[unfolded is_p]
+      show False
+      proof(cases)
+        case (thread_P)
+        moreover from otherwise have "(Cs cs, Th th) \<in> ?A^+"
+            by (unfold rtrancl_eq_or_trancl, auto)
+        ultimately show ?thesis by auto
+      qed
+    qed
+    ultimately show ?thesis by auto
+  qed
+  thus ?thesis by (unfold RAG_es, simp)
+qed
+
+end
+
+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)
-    with ih
-    show ?thesis by (simp add:RAG_create_unchanged)
+    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)
-    with ih show ?thesis by (simp add:RAG_exit_unchanged)
+    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)
-    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
+    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)
+  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)
-    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"
+    interpret vt: valid_trace_p s e th cs using P
+      by (unfold_locales, simp)
+    show ?thesis
     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
+      then interpret vt_h: valid_trace_p_h s e th cs
+        by (unfold_locales, simp)
+      show ?thesis using Cons by (simp add: vt_h.acylic_RAG_kept) 
     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
+      then interpret vt_w: valid_trace_p_w s e th cs
+        by (unfold_locales, simp)
+      show ?thesis using Cons by (simp add: vt_w.acylic_RAG_kept) 
+    qed
+  next
+    case (V th cs)
+    interpret vt: valid_trace_v s e th cs using V
+      by (unfold_locales, simp)
+    show ?thesis
+    proof(cases "vt.rest = []")
+      case True
+      then interpret vt_e: valid_trace_v_e s e th cs
+        by (unfold_locales, simp)
+      show ?thesis by (simp add: Cons.hyps(2) vt_e.acylic_RAG_kept) 
     next
-      case (Set thread prio)
-      with ih
-      thm RAG_set_unchanged
-      show ?thesis by (simp add:RAG_set_unchanged)
+      case False
+      then interpret vt_n: valid_trace_v_n s e th cs
+        by (unfold_locales, simp)
+      show ?thesis by (simp add: Cons.hyps(2) vt_n.acylic_RAG_kept) 
     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)
+    case (Set th prio)
+    interpret vt: valid_trace_set s e th prio using Set
       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)
+    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
+  qed
+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 held_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 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)" .
+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
-  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: 
+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)"
+  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
+
+
+context valid_trace
+begin
+
+lemma finite_subtree_threads:
+    "finite {th'. Th th' \<in> subtree (RAG s) (Th th)}" (is "finite ?A")
+proof -
+  have "?A = the_thread ` {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}"
+        by (auto, insert image_iff, fastforce)
+  moreover have "finite {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}"
+        (is "finite ?B")
+  proof -
+     have "?B = (subtree (RAG s) (Th th)) \<inter> {Th th' | th'. True}"
+      by auto
+     moreover have "... \<subseteq> (subtree (RAG s) (Th th))" by auto
+     moreover have "finite ..." by (simp add: fsbtRAGs.finite_subtree) 
+     ultimately show ?thesis by auto
+  qed
+  ultimately show ?thesis by auto
+qed
+
+lemma le_cp:
+  shows "preced th s \<le> cp s th"
+  proof(unfold cp_alt_def, rule Max_ge)
+    show "finite (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
+      by (simp add: finite_subtree_threads)
+  next
+    show "preced th s \<in> the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}"
+      by (simp add: subtree_def the_preced_def)   
+  qed
+
+lemma cp_le:
+  assumes th_in: "th \<in> threads s"
+  shows "cp s th \<le> Max (the_preced s ` threads s)"
+proof(unfold cp_alt_def, rule Max_f_mono)
+  show "finite (threads s)" by (simp add: finite_threads) 
+next
+  show " {th'. Th th' \<in> subtree (RAG s) (Th th)} \<noteq> {}"
+    using subtree_def by fastforce
+next
+  show "{th'. Th th' \<in> subtree (RAG s) (Th th)} \<subseteq> threads s"
+    using assms
+    by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq 
+        node.inject(1) rtranclD subsetI subtree_def trancl_domain) 
+qed
+
+lemma max_cp_eq: 
+  shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
+  (is "?L = ?R")
+proof -
+  have "?L \<le> ?R" 
+  proof(cases "threads s = {}")
+    case False
+    show ?thesis 
+      by (rule Max.boundedI, 
+          insert cp_le, 
+          auto simp:finite_threads False)
+  qed auto
+  moreover have "?R \<le> ?L"
+    by (rule Max_fg_mono, 
+        simp add: finite_threads,
+        simp add: le_cp the_preced_def)
+  ultimately show ?thesis by auto
+qed
+
+lemma wf_RAG_converse: 
   shows "wf ((RAG s)^-1)"
 proof(rule finite_acyclic_wf_converse)
   from finite_RAG 
@@ -1243,208 +2279,47 @@
   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"
+lemma chain_building:
+  assumes "node \<in> Domain (RAG s)"
+  obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG 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)
+  from assms have "node \<in> Range ((RAG s)^-1)" by auto
+  from wf_base[OF wf_RAG_converse this]
+  obtain b where h_b: "(b, node) \<in> ((RAG s)\<inverse>)\<^sup>+" "\<forall>c. (c, b) \<notin> (RAG s)\<inverse>" by auto
+  obtain th' where eq_b: "b = Th th'"
+  proof(cases b)
+    case (Cs cs)
+    from h_b(1)[unfolded trancl_converse] 
+    have "(node, b) \<in> ((RAG s)\<^sup>+)" by auto
+    from tranclE[OF this]
+    obtain n where "(n, b) \<in> RAG s" by auto
+    from this[unfolded Cs]
+    obtain th1 where "waiting s th1 cs"
+      by (unfold s_RAG_def, fold waiting_eq, auto)
+    from waiting_holding[OF this]
+    obtain th2 where "holding s th2 cs" .
+    hence "(Cs cs, Th th2) \<in> RAG s"
+      by (unfold s_RAG_def, fold holding_eq, auto)
+    with h_b(2)[unfolded Cs, rule_format]
+    have False by auto
+    thus ?thesis by auto
+  qed auto
+  have "th' \<in> readys s" 
+  proof -
+    from h_b(2)[unfolded eq_b]
+    have "\<forall>cs. \<not> waiting s th' cs"
+      by (unfold s_RAG_def, fold waiting_eq, auto)
+    moreover have "th' \<in> threads s"
+    proof(rule rg_RAG_threads)
+      from tranclD[OF h_b(1), unfolded eq_b]
+      obtain z where "(z, Th th') \<in> (RAG s)" by auto
+      thus "Th th' \<in> Range (RAG s)" by auto
     qed
-  next
-    case vt_nil
-    thus ?case by (auto simp:wq_def)
+    ultimately show ?thesis by (auto simp:readys_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
+  moreover have "(node, Th th') \<in> (RAG s)^+" 
+    using h_b(1)[unfolded trancl_converse] eq_b by auto
+  ultimately show ?thesis using that by metis
 qed
 
 text {* \noindent
@@ -1466,182 +2341,6 @@
 
 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 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)"
@@ -1657,7 +2356,39 @@
 lemma count_rec3 [simp]: 
   shows "count Q [] =  0"
   by (unfold count_def, auto)
-  
+
+lemma cntP_simp1[simp]:
+  "cntP (P th cs'#s) th = cntP s th + 1"
+  by (unfold cntP_def, simp)
+
+lemma cntP_simp2[simp]:
+  assumes "th' \<noteq> th"
+  shows "cntP (P th cs'#s) th' = cntP s th'"
+  using assms
+  by (unfold cntP_def, simp)
+
+lemma cntP_simp3[simp]:
+  assumes "\<not> isP e"
+  shows "cntP (e#s) th' = cntP s th'"
+  using assms
+  by (unfold cntP_def, cases e, simp+)
+
+lemma cntV_simp1[simp]:
+  "cntV (V th cs'#s) th = cntV s th + 1"
+  by (unfold cntV_def, simp)
+
+lemma cntV_simp2[simp]:
+  assumes "th' \<noteq> th"
+  shows "cntV (V th cs'#s) th' = cntV s th'"
+  using assms
+  by (unfold cntV_def, simp)
+
+lemma cntV_simp3[simp]:
+  assumes "\<not> isV e"
+  shows "cntV (e#s) th' = cntV s th'"
+  using assms
+  by (unfold cntV_def, cases e, simp+)
+
 lemma cntP_diff_inv:
   assumes "cntP (e#s) th \<noteq> cntP s th"
   shows "isP e \<and> actor e = th"
@@ -1667,17 +2398,7 @@
   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 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) (* ccc *)
-
+  
 lemma cntV_diff_inv:
   assumes "cntV (e#s) th \<noteq> cntV s th"
   shows "isV e \<and> actor e = th"
@@ -1688,659 +2409,1256 @@
         insert assms V, auto simp:cntV_def)
 qed (insert assms, auto simp:cntV_def)
 
+lemma children_RAG_alt_def:
+  "children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}"
+  by (unfold s_RAG_def, auto simp:children_def holding_eq)
+
+lemma holdents_alt_def:
+  "holdents s th = the_cs ` (children (RAG (s::state)) (Th th))"
+  by (unfold children_RAG_alt_def holdents_def, simp add: image_image)
+
+lemma cntCS_alt_def:
+  "cntCS s th = card (children (RAG s) (Th th))"
+  apply (unfold children_RAG_alt_def cntCS_def holdents_def)
+  by (rule card_image[symmetric], auto simp:inj_on_def)
+
 context valid_trace
 begin
 
-text {* (* ddd *) \noindent
-  The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} 
-  of one particular thread. 
-*} 
-
-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)"
+lemma finite_holdents: "finite (holdents s th)"
+  by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)
+  
+end
+
+context valid_trace_p_w
+begin
+
+lemma holding_s_holder: "holding s holder cs"
+  by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
+
+lemma holding_es_holder: "holding (e#s) holder cs"
+  by (unfold s_holding_def, fold wq_def, unfold wq_es_cs wq_s_cs, auto)
+
+lemma holdents_es:
+  shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") 
 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"
+  { fix cs'
+    assume "cs' \<in> ?L"
+    hence h: "holding (e#s) th' cs'" by (auto simp:holdents_def)
+    have "holding s th' cs'"
+    proof(cases "cs' = cs")
+      case True
+      from held_unique[OF h[unfolded True] holding_es_holder]
+      have "th' = holder" .
+      thus ?thesis 
+        by (unfold True holdents_def, insert holding_s_holder, simp)
+    next
+      case False
+      hence "wq (e#s) cs' = wq s cs'" by simp
+      from h[unfolded s_holding_def, folded wq_def, unfolded this]
+      show ?thesis
+       by (unfold s_holding_def, fold wq_def, auto)
+    qed 
+    hence "cs' \<in> ?R" by (auto simp:holdents_def)
+  } moreover {
+    fix cs'
+    assume "cs' \<in> ?R"
+    hence h: "holding s th' cs'" by (auto simp:holdents_def)
+    have "holding (e#s) th' cs'"
+    proof(cases "cs' = cs")
+      case True
+      from held_unique[OF h[unfolded True] holding_s_holder]
+      have "th' = holder" .
+      thus ?thesis 
+        by (unfold True holdents_def, insert holding_es_holder, simp)
+    next
+      case False
+      hence "wq s cs' = wq (e#s) cs'" by simp
+      from h[unfolded s_holding_def, folded wq_def, unfolded this]
       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
+       by (unfold s_holding_def, fold wq_def, auto)
+    qed 
+    hence "cs' \<in> ?L" by (auto simp:holdents_def)
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_es_th[simp]: "cntCS (e#s) th' = cntCS s th'"
+ by (unfold cntCS_def holdents_es, simp)
+
+lemma th_not_ready_es: 
+  shows "th \<notin> readys (e#s)"
+  using waiting_es_th_cs 
+  by (unfold readys_def, auto)
+
+end
+  
+context valid_trace_p_h
+begin
+
+lemma th_not_waiting':
+  "\<not> waiting (e#s) th cs'"
+proof(cases "cs' = cs")
+  case True
+  show ?thesis
+    by (unfold True s_waiting_def, fold wq_def, unfold wq_es_cs', auto)
+next
+  case False
+  from th_not_waiting[of cs', unfolded s_waiting_def, folded wq_def]
+  show ?thesis
+    by (unfold s_waiting_def, fold wq_def, insert False, simp)
+qed
+
+lemma ready_th_es: 
+  shows "th \<in> readys (e#s)"
+  using th_not_waiting'
+  by (unfold readys_def, insert live_th_es, auto)
+
+lemma holdents_es_th:
+  "holdents (e#s) th = (holdents s th) \<union> {cs}" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume "cs' \<in> ?L" 
+    hence "holding (e#s) th cs'"
+      by (unfold holdents_def, auto)
+    hence "cs' \<in> ?R"
+     by (cases rule:holding_esE, auto simp:holdents_def)
+  } moreover {
+    fix cs'
+    assume "cs' \<in> ?R"
+    hence "holding s th cs' \<or> cs' = cs" 
+      by (auto simp:holdents_def)
+    hence "cs' \<in> ?L"
+    proof
+      assume "holding s th cs'"
+      from holding_kept[OF this]
+      show ?thesis by (auto simp:holdents_def)
     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
+      assume "cs' = cs"
+      thus ?thesis using holding_es_th_cs
+        by (unfold holdents_def, auto)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_es_th: "cntCS (e#s) th = cntCS s th + 1"
+proof -
+  have "card (holdents s th \<union> {cs}) = card (holdents s th) + 1"
+  proof(subst card_Un_disjoint)
+    show "holdents s th \<inter> {cs} = {}"
+      using not_holding_s_th_cs by (auto simp:holdents_def)
+  qed (auto simp:finite_holdents)
+  thus ?thesis
+   by (unfold cntCS_def holdents_es_th, simp)
+qed
+
+lemma no_holder: 
+  "\<not> holding s th' cs"
+proof
+  assume otherwise: "holding s th' cs"
+  from this[unfolded s_holding_def, folded wq_def, unfolded we]
+  show False by auto
+qed
+
+lemma holdents_es_th':
+  assumes "th' \<noteq> th"
+  shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume "cs' \<in> ?L"
+    hence h_e: "holding (e#s) th' cs'" by (auto simp:holdents_def)
+    have "cs' \<noteq> cs"
+    proof
+      assume "cs' = cs"
+      from held_unique[OF h_e[unfolded this] holding_es_th_cs]
+      have "th' = th" .
+      with assms show False by simp
+    qed
+    from h_e[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp[OF this]]
+    have "th' \<in> set (wq s cs') \<and> th' = hd (wq s cs')" .
+    hence "cs' \<in> ?R" 
+      by (unfold holdents_def s_holding_def, fold wq_def, auto)
+  } moreover {
+    fix cs'
+    assume "cs' \<in> ?R"
+    hence "holding s th' cs'" by (auto simp:holdents_def)
+    from holding_kept[OF this]
+    have "holding (e # s) th' cs'" .
+    hence "cs' \<in> ?L"
+      by (unfold holdents_def, auto)
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_es_th'[simp]: 
+  assumes "th' \<noteq> th"
+  shows "cntCS (e#s) th' = cntCS s th'"
+  by (unfold cntCS_def holdents_es_th'[OF assms], simp)
+
+end
+
+context valid_trace_p
+begin
+
+lemma readys_kept1: 
+  assumes "th' \<noteq> th"
+  and "th' \<in> readys (e#s)"
+  shows "th' \<in> readys s"
+proof -
+  { fix cs'
+    assume wait: "waiting s th' cs'"
+    have n_wait: "\<not> waiting (e#s) th' cs'" 
+        using assms(2)[unfolded readys_def] by auto
+    have False
+    proof(cases "cs' = cs")
+      case False
+      with n_wait wait
+      show ?thesis 
+        by (unfold s_waiting_def, fold wq_def, auto)
     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)
+      case True
+      show ?thesis
+      proof(cases "wq s cs = []")
+        case True
+        then interpret vt: valid_trace_p_h
+          by (unfold_locales, simp)
+        show ?thesis using n_wait wait waiting_kept by auto 
+      next
+        case False
+        then interpret vt: valid_trace_p_w by (unfold_locales, simp)
+        show ?thesis using n_wait wait waiting_kept by blast 
+      qed
+    qed
+  } with assms(2) show ?thesis  
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_kept2: 
+  assumes "th' \<noteq> th"
+  and "th' \<in> readys s"
+  shows "th' \<in> readys (e#s)"
+proof -
+  { fix cs'
+    assume wait: "waiting (e#s) th' cs'"
+    have n_wait: "\<not> waiting s th' cs'" 
+        using assms(2)[unfolded readys_def] by auto
+    have False
+    proof(cases "cs' = cs")
+      case False
+      with n_wait wait
       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
+        by (unfold s_waiting_def, fold wq_def, auto)
+    next
+      case True
+      show ?thesis
+      proof(cases "wq s cs = []")
+        case True
+        then interpret vt: valid_trace_p_h
+          by (unfold_locales, simp)
+        show ?thesis using n_wait vt.waiting_esE wait by blast 
+      next
+        case False
+        then interpret vt: valid_trace_p_w by (unfold_locales, simp)
+        show ?thesis using assms(1) n_wait vt.waiting_esE wait by auto 
       qed
+    qed
+  } with assms(2) show ?thesis  
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_simp [simp]:
+  assumes "th' \<noteq> th"
+  shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+  using readys_kept1[OF assms] readys_kept2[OF assms]
+  by metis
+
+lemma cnp_cnv_cncs_kept: (* ddd *)
+  assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+  shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof(cases "th' = th")
+  case True
+  note eq_th' = this
+  show ?thesis
+  proof(cases "wq s cs = []")
+    case True
+    then interpret vt: valid_trace_p_h by (unfold_locales, simp)
+    show ?thesis
+      using assms eq_th' is_p ready_th_s vt.cntCS_es_th vt.ready_th_es pvD_def by auto 
+  next
+    case False
+    then interpret vt: valid_trace_p_w by (unfold_locales, simp)
+    show ?thesis
+      using add.commute add.left_commute assms eq_th' is_p live_th_s 
+            ready_th_s vt.th_not_ready_es pvD_def
+      apply (auto)
+      by (fold is_p, simp)
+  qed
+next
+  case False
+  note h_False = False
+  thus ?thesis
+  proof(cases "wq s cs = []")
+    case True
+    then interpret vt: valid_trace_p_h by (unfold_locales, simp)
+    show ?thesis using assms
+      by (insert True h_False pvD_def, auto split:if_splits,unfold is_p, auto)
+  next
+    case False
+    then interpret vt: valid_trace_p_w by (unfold_locales, simp)
+    show ?thesis using assms
+      by (insert False h_False pvD_def, auto split:if_splits,unfold is_p, auto)
+  qed
+qed
+
+end
+
+
+context valid_trace_v (* ccc *)
+begin
+
+lemma holding_th_cs_s: 
+  "holding s th cs" 
+ by  (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
+
+lemma th_ready_s [simp]: "th \<in> readys s"
+  using runing_th_s
+  by (unfold runing_def readys_def, auto)
+
+lemma th_live_s [simp]: "th \<in> threads s"
+  using th_ready_s by (unfold readys_def, auto)
+
+lemma th_ready_es [simp]: "th \<in> readys (e#s)"
+  using runing_th_s neq_t_th
+  by (unfold is_v runing_def readys_def, auto)
+
+lemma th_live_es [simp]: "th \<in> threads (e#s)"
+  using th_ready_es by (unfold readys_def, auto)
+
+lemma pvD_th_s[simp]: "pvD s th = 0"
+  by (unfold pvD_def, simp)
+
+lemma pvD_th_es[simp]: "pvD (e#s) th = 0"
+  by (unfold pvD_def, simp)
+
+lemma cntCS_s_th [simp]: "cntCS s th > 0"
+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
+    by simp
+  ultimately show ?thesis
+    by (unfold cntCS_def, 
+        auto intro!:card_gt_0_iff[symmetric, THEN iffD1])
+qed
+
+end
+
+context valid_trace_v_n
+begin
+
+lemma not_ready_taker_s[simp]: 
+  "taker \<notin> readys s"
+  using waiting_taker
+  by (unfold readys_def, auto)
+
+lemma taker_live_s [simp]: "taker \<in> threads s"
+proof -
+  have "taker \<in> set wq'" by (simp add: eq_wq') 
+  from th'_in_inv[OF this]
+  have "taker \<in> set rest" .
+  hence "taker \<in> set (wq s cs)" by (simp add: wq_s_cs) 
+  thus ?thesis using wq_threads by auto 
+qed
+
+lemma taker_live_es [simp]: "taker \<in> threads (e#s)"
+  using taker_live_s threads_es by blast
+
+lemma taker_ready_es [simp]:
+  shows "taker \<in> readys (e#s)"
+proof -
+  { fix cs'
+    assume "waiting (e#s) taker cs'"
+    hence False
+    proof(cases rule:waiting_esE)
+      case 1
+      thus ?thesis using waiting_taker waiting_unique by auto 
+    qed simp
+  } thus ?thesis by (unfold readys_def, auto)
+qed
+
+lemma neq_taker_th: "taker \<noteq> th"
+  using th_not_waiting waiting_taker by blast
+
+lemma not_holding_taker_s_cs:
+  shows "\<not> holding s taker cs"
+  using holding_cs_eq_th neq_taker_th by auto
+
+lemma holdents_es_taker:
+  "holdents (e#s) taker = holdents s taker \<union> {cs}" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume "cs' \<in> ?L"
+    hence "holding (e#s) taker cs'" by (auto simp:holdents_def)
+    hence "cs' \<in> ?R"
+    proof(cases rule:holding_esE)
+      case 2
+      thus ?thesis by (auto simp:holdents_def)
+    qed auto
+  } moreover {
+    fix cs'
+    assume "cs' \<in> ?R"
+    hence "holding s taker cs' \<or> cs' = cs" by (auto simp:holdents_def)
+    hence "cs' \<in> ?L" 
+    proof
+      assume "holding s taker cs'"
+      hence "holding (e#s) taker cs'" 
+          using holding_esI2 holding_taker by fastforce 
+      thus ?thesis by (auto simp:holdents_def)
     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
+      assume "cs' = cs"
+      with holding_taker
+      show ?thesis by (auto simp:holdents_def)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_es_taker [simp]: "cntCS (e#s) taker = cntCS s taker + 1"
+proof -
+  have "card (holdents s taker \<union> {cs}) = card (holdents s taker) + 1"
+  proof(subst card_Un_disjoint)
+    show "holdents s taker \<inter> {cs} = {}"
+      using not_holding_taker_s_cs by (auto simp:holdents_def)
+  qed (auto simp:finite_holdents)
+  thus ?thesis 
+    by (unfold cntCS_def, insert holdents_es_taker, simp)
+qed
+
+lemma pvD_taker_s[simp]: "pvD s taker = 1"
+  by (unfold pvD_def, simp)
+
+lemma pvD_taker_es[simp]: "pvD (e#s) taker = 0"
+  by (unfold pvD_def, simp)  
+
+lemma pvD_th_s[simp]: "pvD s th = 0"
+  by (unfold pvD_def, simp)
+
+lemma pvD_th_es[simp]: "pvD (e#s) th = 0"
+  by (unfold pvD_def, simp)
+
+lemma holdents_es_th:
+  "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume "cs' \<in> ?L"
+    hence "holding (e#s) th cs'" by (auto simp:holdents_def)
+    hence "cs' \<in> ?R"
+    proof(cases rule:holding_esE)
+      case 2
+      thus ?thesis by (auto simp:holdents_def)
+    qed (insert neq_taker_th, auto)
+  } moreover {
+    fix cs'
+    assume "cs' \<in> ?R"
+    hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def)
+    from holding_esI2[OF this]
+    have "cs' \<in> ?L" by (auto simp:holdents_def)
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1"
+proof -
+  have "card (holdents s th - {cs}) = card (holdents s th) - 1"
+  proof -
+    have "cs \<in> holdents s th" using holding_th_cs_s
+      by (auto simp:holdents_def)
+    moreover have "finite (holdents s th)"
+        by (simp add: finite_holdents) 
+    ultimately show ?thesis by auto
+  qed
+  thus ?thesis by (unfold cntCS_def holdents_es_th)
+qed
+
+lemma holdents_kept:
+  assumes "th' \<noteq> taker"
+  and "th' \<noteq> th"
+  shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume h: "cs' \<in> ?L"
+    have "cs' \<in> ?R"
+    proof(cases "cs' = cs")
+      case False
+      hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
+      from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
+      from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
+      show ?thesis
+        by (unfold holdents_def s_holding_def, fold wq_def, auto)
+    next
+      case True
+      from h[unfolded this]
+      have "holding (e#s) th' cs" by (auto simp:holdents_def)
+      from held_unique[OF this holding_taker]
+      have "th' = taker" .
+      with assms show ?thesis by auto
+    qed
+  } moreover {
+    fix cs'
+    assume h: "cs' \<in> ?R"
+    have "cs' \<in> ?L"
+    proof(cases "cs' = cs")
+      case False
+      hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
+      from h have "holding s th' cs'" by (auto simp:holdents_def)
+      from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
       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
+        by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
+    next
+      case True
+      from h[unfolded this]
+      have "holding s th' cs" by (auto simp:holdents_def)
+      from held_unique[OF this holding_th_cs_s]
+      have "th' = th" .
+      with assms show ?thesis by auto
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_kept [simp]:
+  assumes "th' \<noteq> taker"
+  and "th' \<noteq> th"
+  shows "cntCS (e#s) th' = cntCS s th'"
+  by (unfold cntCS_def holdents_kept[OF assms], simp)
+
+lemma readys_kept1: 
+  assumes "th' \<noteq> taker"
+  and "th' \<in> readys (e#s)"
+  shows "th' \<in> readys s"
+proof -
+  { fix cs'
+    assume wait: "waiting s th' cs'"
+    have n_wait: "\<not> waiting (e#s) th' cs'" 
+        using assms(2)[unfolded readys_def] by auto
+    have False
+    proof(cases "cs' = cs")
+      case False
+      with n_wait wait
+      show ?thesis 
+        by (unfold s_waiting_def, fold wq_def, auto)
+    next
+      case True
+      have "th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)" 
+        using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
+      moreover have "\<not> (th' \<in> set rest \<and> th' \<noteq> hd (taker # rest'))" 
+        using n_wait[unfolded True s_waiting_def, folded wq_def, 
+                    unfolded wq_es_cs set_wq', unfolded eq_wq'] .
+      ultimately have "th' = taker" by auto
+      with assms(1)
+      show ?thesis by simp
+    qed
+  } with assms(2) show ?thesis  
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_kept2: 
+  assumes "th' \<noteq> taker"
+  and "th' \<in> readys s"
+  shows "th' \<in> readys (e#s)"
+proof -
+  { fix cs'
+    assume wait: "waiting (e#s) th' cs'"
+    have n_wait: "\<not> waiting s th' cs'" 
+        using assms(2)[unfolded readys_def] by auto
+    have False
+    proof(cases "cs' = cs")
+      case False
+      with n_wait wait
+      show ?thesis 
+        by (unfold s_waiting_def, fold wq_def, auto)
+    next
+      case True
+      have "th' \<in> set rest \<and> th' \<noteq> hd (taker # rest')"
+          using  wait [unfolded True s_waiting_def, folded wq_def, 
+                    unfolded wq_es_cs set_wq', unfolded eq_wq']  .
+      moreover have "\<not> (th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest))"
+          using n_wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
+      ultimately have "th' = taker" by auto
+      with assms(1)
+      show ?thesis by simp
+    qed
+  } with assms(2) show ?thesis  
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_simp [simp]:
+  assumes "th' \<noteq> taker"
+  shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+  using readys_kept1[OF assms] readys_kept2[OF assms]
+  by metis
+
+lemma cnp_cnv_cncs_kept:
+  assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+  shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof -
+  { assume eq_th': "th' = taker"
+    have ?thesis
+      apply (unfold eq_th' pvD_taker_es cntCS_es_taker)
+      by (insert neq_taker_th assms[unfolded eq_th'], unfold is_v, simp)
+  } moreover {
+    assume eq_th': "th' = th"
+    have ?thesis 
+      apply (unfold eq_th' pvD_th_es cntCS_es_th)
+      by (insert assms[unfolded eq_th'], unfold is_v, simp)
+  } moreover {
+    assume h: "th' \<noteq> taker" "th' \<noteq> th"
+    have ?thesis using assms
+      apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp)
+      by (fold is_v, unfold pvD_def, simp)
+  } ultimately show ?thesis by metis
+qed
+
+end
+
+context valid_trace_v_e
+begin
+
+lemma holdents_es_th:
+  "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume "cs' \<in> ?L"
+    hence "holding (e#s) th cs'" by (auto simp:holdents_def)
+    hence "cs' \<in> ?R"
+    proof(cases rule:holding_esE)
+      case 1
+      thus ?thesis by (auto simp:holdents_def)
+    qed 
+  } moreover {
+    fix cs'
+    assume "cs' \<in> ?R"
+    hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def)
+    from holding_esI2[OF this]
+    have "cs' \<in> ?L" by (auto simp:holdents_def)
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1"
+proof -
+  have "card (holdents s th - {cs}) = card (holdents s th) - 1"
+  proof -
+    have "cs \<in> holdents s th" using holding_th_cs_s
+      by (auto simp:holdents_def)
+    moreover have "finite (holdents s th)"
+        by (simp add: finite_holdents) 
+    ultimately show ?thesis by auto
+  qed
+  thus ?thesis by (unfold cntCS_def holdents_es_th)
+qed
+
+lemma holdents_kept:
+  assumes "th' \<noteq> th"
+  shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume h: "cs' \<in> ?L"
+    have "cs' \<in> ?R"
+    proof(cases "cs' = cs")
+      case False
+      hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
+      from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
+      from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
+      show ?thesis
+        by (unfold holdents_def s_holding_def, fold wq_def, auto)
+    next
+      case True
+      from h[unfolded this]
+      have "holding (e#s) th' cs" by (auto simp:holdents_def)
+      from this[unfolded s_holding_def, folded wq_def, 
+            unfolded wq_es_cs nil_wq']
+      show ?thesis by auto
+    qed
+  } moreover {
+    fix cs'
+    assume h: "cs' \<in> ?R"
+    have "cs' \<in> ?L"
+    proof(cases "cs' = cs")
+      case False
+      hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
+      from h have "holding s th' cs'" by (auto simp:holdents_def)
+      from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
+      show ?thesis
+        by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
+    next
+      case True
+      from h[unfolded this]
+      have "holding s th' cs" by (auto simp:holdents_def)
+      from held_unique[OF this holding_th_cs_s]
+      have "th' = th" .
+      with assms show ?thesis by auto
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_kept [simp]:
+  assumes "th' \<noteq> th"
+  shows "cntCS (e#s) th' = cntCS s th'"
+  by (unfold cntCS_def holdents_kept[OF assms], simp)
+
+lemma readys_kept1: 
+  assumes "th' \<in> readys (e#s)"
+  shows "th' \<in> readys s"
+proof -
+  { fix cs'
+    assume wait: "waiting s th' cs'"
+    have n_wait: "\<not> waiting (e#s) th' cs'" 
+        using assms(1)[unfolded readys_def] by auto
+    have False
+    proof(cases "cs' = cs")
+      case False
+      with n_wait wait
+      show ?thesis 
+        by (unfold s_waiting_def, fold wq_def, auto)
+    next
+      case True
+      have "th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)" 
+        using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . 
+      hence "th' \<in> set rest" by auto
+      with set_wq' have "th' \<in> set wq'" by metis
+      with nil_wq' show ?thesis by simp
+    qed
+  } thus ?thesis using assms
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_kept2: 
+  assumes "th' \<in> readys s"
+  shows "th' \<in> readys (e#s)"
+proof -
+  { fix cs'
+    assume wait: "waiting (e#s) th' cs'"
+    have n_wait: "\<not> waiting s th' cs'" 
+        using assms[unfolded readys_def] by auto
+    have False
+    proof(cases "cs' = cs")
+      case False
+      with n_wait wait
+      show ?thesis 
+        by (unfold s_waiting_def, fold wq_def, auto)
     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   
+      case True
+      have "th' \<in> set [] \<and> th' \<noteq> hd []"
+        using wait[unfolded True s_waiting_def, folded wq_def, 
+              unfolded wq_es_cs nil_wq'] .
+      thus ?thesis by simp
     qed
+  } with assms show ?thesis  
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_simp [simp]:
+  shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+  using readys_kept1[OF assms] readys_kept2[OF assms]
+  by metis
+
+lemma cnp_cnv_cncs_kept:
+  assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+  shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof -
+  {
+    assume eq_th': "th' = th"
+    have ?thesis 
+      apply (unfold eq_th' pvD_th_es cntCS_es_th)
+      by (insert assms[unfolded eq_th'], unfold is_v, simp)
+  } moreover {
+    assume h: "th' \<noteq> th"
+    have ?thesis using assms
+      apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp)
+      by (fold is_v, unfold pvD_def, simp)
+  } ultimately show ?thesis by metis
+qed
+
+end
+
+context valid_trace_v
+begin
+
+lemma cnp_cnv_cncs_kept:
+  assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+  shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof(cases "rest = []")
+  case True
+  then interpret vt: valid_trace_v_e by (unfold_locales, simp)
+  show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast 
+next
+  case False
+  then interpret vt: valid_trace_v_n by (unfold_locales, simp)
+  show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast 
+qed
+
+end
+
+context valid_trace_create
+begin
+
+lemma th_not_live_s [simp]: "th \<notin> threads s"
+proof -
+  from pip_e[unfolded is_create]
+  show ?thesis by (cases, simp)
+qed
+
+lemma th_not_ready_s [simp]: "th \<notin> readys s"
+  using th_not_live_s by (unfold readys_def, simp)
+
+lemma th_live_es [simp]: "th \<in> threads (e#s)"
+  by (unfold is_create, simp)
+
+lemma not_waiting_th_s [simp]: "\<not> waiting s th cs'"
+proof
+  assume "waiting s th cs'"
+  from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+  have "th \<in> set (wq s cs')" by auto
+  from wq_threads[OF this] have "th \<in> threads s" .
+  with th_not_live_s show False by simp
+qed
+
+lemma not_holding_th_s [simp]: "\<not> holding s th cs'"
+proof
+  assume "holding s th cs'"
+  from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
+  have "th \<in> set (wq s cs')" by auto
+  from wq_threads[OF this] have "th \<in> threads s" .
+  with th_not_live_s show False by simp
+qed
+
+lemma not_waiting_th_es [simp]: "\<not> waiting (e#s) th cs'"
+proof
+  assume "waiting (e # s) th cs'"
+  from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+  have "th \<in> set (wq s cs')" by auto
+  from wq_threads[OF this] have "th \<in> threads s" .
+  with th_not_live_s show False by simp
+qed
+
+lemma not_holding_th_es [simp]: "\<not> holding (e#s) th cs'"
+proof
+  assume "holding (e # s) th cs'"
+  from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
+  have "th \<in> set (wq s cs')" by auto
+  from wq_threads[OF this] have "th \<in> threads s" .
+  with th_not_live_s show False by simp
+qed
+
+lemma ready_th_es [simp]: "th \<in> readys (e#s)"
+  by (simp add:readys_def)
+
+lemma holdents_th_s: "holdents s th = {}"
+  by (unfold holdents_def, auto)
+
+lemma holdents_th_es: "holdents (e#s) th = {}"
+  by (unfold holdents_def, auto)
+
+lemma cntCS_th_s [simp]: "cntCS s th = 0"
+  by (unfold cntCS_def, simp add:holdents_th_s)
+
+lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0"
+  by (unfold cntCS_def, simp add:holdents_th_es)
+
+lemma pvD_th_s [simp]: "pvD s th = 0"
+  by (unfold pvD_def, simp)
+
+lemma pvD_th_es [simp]: "pvD (e#s) th = 0"
+  by (unfold pvD_def, simp)
+
+lemma holdents_kept:
+  assumes "th' \<noteq> th"
+  shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume h: "cs' \<in> ?L"
+    hence "cs' \<in> ?R"
+      by (unfold holdents_def s_holding_def, fold wq_def, 
+             unfold wq_neq_simp, auto)
+  } moreover {
+    fix cs'
+    assume h: "cs' \<in> ?R"
+    hence "cs' \<in> ?L"
+      by (unfold holdents_def s_holding_def, fold wq_def, 
+             unfold wq_neq_simp, auto)
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_kept [simp]:
+  assumes "th' \<noteq> th"
+  shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
+  using holdents_kept[OF assms]
+  by (unfold cntCS_def, simp)
+
+lemma readys_kept1: 
+  assumes "th' \<noteq> th"
+  and "th' \<in> readys (e#s)"
+  shows "th' \<in> readys s"
+proof -
+  { fix cs'
+    assume wait: "waiting s th' cs'"
+    have n_wait: "\<not> waiting (e#s) th' cs'" 
+      using assms by (auto simp:readys_def)
+    from wait[unfolded s_waiting_def, folded wq_def]
+         n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+    have False by auto
+  } thus ?thesis using assms
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_kept2: 
+  assumes "th' \<noteq> th"
+  and "th' \<in> readys s"
+  shows "th' \<in> readys (e#s)"
+proof -
+  { fix cs'
+    assume wait: "waiting (e#s) th' cs'"
+    have n_wait: "\<not> waiting s th' cs'"
+      using assms(2) by (auto simp:readys_def)
+    from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+         n_wait[unfolded s_waiting_def, folded wq_def]
+    have False by auto
+  } with assms show ?thesis  
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_simp [simp]:
+  assumes "th' \<noteq> th"
+  shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+  using readys_kept1[OF assms] readys_kept2[OF assms]
+  by metis
+
+lemma pvD_kept [simp]:
+  assumes "th' \<noteq> th"
+  shows "pvD (e#s) th' = pvD s th'"
+  using assms
+  by (unfold pvD_def, simp)
+
+lemma cnp_cnv_cncs_kept:
+  assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+  shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof -
+  {
+    assume eq_th': "th' = th"
+    have ?thesis using assms
+      by (unfold eq_th', simp, unfold is_create, simp)
+  } moreover {
+    assume h: "th' \<noteq> th"
+    hence ?thesis using assms
+      by (simp, simp add:is_create)
+  } ultimately show ?thesis by metis
+qed
+
+end
+
+context valid_trace_exit
+begin
+
+lemma th_live_s [simp]: "th \<in> threads s"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis
+  by (cases, unfold runing_def readys_def, simp)
+qed
+
+lemma th_ready_s [simp]: "th \<in> readys s"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis
+  by (cases, unfold runing_def, simp)
+qed
+
+lemma th_not_live_es [simp]: "th \<notin> threads (e#s)"
+  by (unfold is_exit, simp)
+
+lemma not_holding_th_s [simp]: "\<not> holding s th cs'"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis 
+   by (cases, unfold holdents_def, auto)
+qed
+
+lemma cntCS_th_s [simp]: "cntCS s th = 0"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis 
+   by (cases, unfold cntCS_def, simp)
+qed
+
+lemma not_holding_th_es [simp]: "\<not> holding (e#s) th cs'"
+proof
+  assume "holding (e # s) th cs'"
+  from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
+  have "holding s th cs'" 
+    by (unfold s_holding_def, fold wq_def, auto)
+  with not_holding_th_s 
+  show False by simp
+qed
+
+lemma ready_th_es [simp]: "th \<notin> readys (e#s)"
+  by (simp add:readys_def)
+
+lemma holdents_th_s: "holdents s th = {}"
+  by (unfold holdents_def, auto)
+
+lemma holdents_th_es: "holdents (e#s) th = {}"
+  by (unfold holdents_def, auto)
+
+lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0"
+  by (unfold cntCS_def, simp add:holdents_th_es)
+
+lemma pvD_th_s [simp]: "pvD s th = 0"
+  by (unfold pvD_def, simp)
+
+lemma pvD_th_es [simp]: "pvD (e#s) th = 0"
+  by (unfold pvD_def, simp)
+
+lemma holdents_kept:
+  assumes "th' \<noteq> th"
+  shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume h: "cs' \<in> ?L"
+    hence "cs' \<in> ?R"
+      by (unfold holdents_def s_holding_def, fold wq_def, 
+             unfold wq_neq_simp, auto)
+  } moreover {
+    fix cs'
+    assume h: "cs' \<in> ?R"
+    hence "cs' \<in> ?L"
+      by (unfold holdents_def s_holding_def, fold wq_def, 
+             unfold wq_neq_simp, auto)
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_kept [simp]:
+  assumes "th' \<noteq> th"
+  shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
+  using holdents_kept[OF assms]
+  by (unfold cntCS_def, simp)
+
+lemma readys_kept1: 
+  assumes "th' \<noteq> th"
+  and "th' \<in> readys (e#s)"
+  shows "th' \<in> readys s"
+proof -
+  { fix cs'
+    assume wait: "waiting s th' cs'"
+    have n_wait: "\<not> waiting (e#s) th' cs'" 
+      using assms by (auto simp:readys_def)
+    from wait[unfolded s_waiting_def, folded wq_def]
+         n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+    have False by auto
+  } thus ?thesis using assms
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_kept2: 
+  assumes "th' \<noteq> th"
+  and "th' \<in> readys s"
+  shows "th' \<in> readys (e#s)"
+proof -
+  { fix cs'
+    assume wait: "waiting (e#s) th' cs'"
+    have n_wait: "\<not> waiting s th' cs'"
+      using assms(2) by (auto simp:readys_def)
+    from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+         n_wait[unfolded s_waiting_def, folded wq_def]
+    have False by auto
+  } with assms show ?thesis  
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_simp [simp]:
+  assumes "th' \<noteq> th"
+  shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+  using readys_kept1[OF assms] readys_kept2[OF assms]
+  by metis
+
+lemma pvD_kept [simp]:
+  assumes "th' \<noteq> th"
+  shows "pvD (e#s) th' = pvD s th'"
+  using assms
+  by (unfold pvD_def, simp)
+
+lemma cnp_cnv_cncs_kept:
+  assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+  shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof -
+  {
+    assume eq_th': "th' = th"
+    have ?thesis using assms
+      by (unfold eq_th', simp, unfold is_exit, simp)
+  } moreover {
+    assume h: "th' \<noteq> th"
+    hence ?thesis using assms
+      by (simp, simp add:is_exit)
+  } ultimately show ?thesis by metis
+qed
+
+end
+
+context valid_trace_set
+begin
+
+lemma th_live_s [simp]: "th \<in> threads s"
+proof -
+  from pip_e[unfolded is_set]
+  show ?thesis
+  by (cases, unfold runing_def readys_def, simp)
+qed
+
+lemma th_ready_s [simp]: "th \<in> readys s"
+proof -
+  from pip_e[unfolded is_set]
+  show ?thesis
+  by (cases, unfold runing_def, simp)
+qed
+
+lemma th_not_live_es [simp]: "th \<in> threads (e#s)"
+  by (unfold is_set, simp)
+
+
+lemma holdents_kept:
+  shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+  { fix cs'
+    assume h: "cs' \<in> ?L"
+    hence "cs' \<in> ?R"
+      by (unfold holdents_def s_holding_def, fold wq_def, 
+             unfold wq_neq_simp, auto)
+  } moreover {
+    fix cs'
+    assume h: "cs' \<in> ?R"
+    hence "cs' \<in> ?L"
+      by (unfold holdents_def s_holding_def, fold wq_def, 
+             unfold wq_neq_simp, auto)
+  } ultimately show ?thesis by auto
+qed
+
+lemma cntCS_kept [simp]:
+  shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
+  using holdents_kept
+  by (unfold cntCS_def, simp)
+
+lemma threads_kept[simp]:
+  "threads (e#s) = threads s"
+  by (unfold is_set, simp)
+
+lemma readys_kept1: 
+  assumes "th' \<in> readys (e#s)"
+  shows "th' \<in> readys s"
+proof -
+  { fix cs'
+    assume wait: "waiting s th' cs'"
+    have n_wait: "\<not> waiting (e#s) th' cs'" 
+      using assms by (auto simp:readys_def)
+    from wait[unfolded s_waiting_def, folded wq_def]
+         n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+    have False by auto
+  } moreover have "th' \<in> threads s" 
+    using assms[unfolded readys_def] by auto
+  ultimately show ?thesis 
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_kept2: 
+  assumes "th' \<in> readys s"
+  shows "th' \<in> readys (e#s)"
+proof -
+  { fix cs'
+    assume wait: "waiting (e#s) th' cs'"
+    have n_wait: "\<not> waiting s th' cs'"
+      using assms by (auto simp:readys_def)
+    from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+         n_wait[unfolded s_waiting_def, folded wq_def]
+    have False by auto
+  } with assms show ?thesis  
+    by (unfold readys_def, auto)
+qed
+
+lemma readys_simp [simp]:
+  shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+  using readys_kept1 readys_kept2
+  by metis
+
+lemma pvD_kept [simp]:
+  shows "pvD (e#s) th' = pvD s th'"
+  by (unfold pvD_def, simp)
+
+lemma cnp_cnv_cncs_kept:
+  assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+  shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+  using assms
+  by (unfold is_set, simp, fold is_set, simp)
+
+end
+
+context valid_trace
+begin
+
+lemma cnp_cnv_cncs: "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+proof(induct rule:ind)
+  case Nil
+  thus ?case 
+    by (unfold cntP_def cntV_def pvD_def cntCS_def holdents_def 
+              s_holding_def, simp)
+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_create: valid_trace_create s e th prio 
+      using Create by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_create.cnp_cnv_cncs_kept) 
   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)
+    case (Exit th)
+    interpret vt_exit: valid_trace_exit s e th  
+        using Exit by (unfold_locales, simp)
+   show ?thesis using Cons by (simp add: vt_exit.cnp_cnv_cncs_kept) 
+  next
+    case (P th cs)
+    interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_p.cnp_cnv_cncs_kept) 
+  next
+    case (V th cs)
+    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_v.cnp_cnv_cncs_kept) 
+  next
+    case (Set th prio)
+    interpret vt_set: valid_trace_set s e th prio
+        using Set by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_set.cnp_cnv_cncs_kept) 
   qed
 qed
 
+lemma not_thread_holdents:
+  assumes not_in: "th \<notin> threads s" 
+  shows "holdents s th = {}"
+proof -
+  { fix cs
+    assume "cs \<in> holdents s th"
+    hence "holding s th cs" by (auto simp:holdents_def)
+    from this[unfolded s_holding_def, folded wq_def]
+    have "th \<in> set (wq s cs)" by auto
+    with wq_threads have "th \<in> threads s" by auto
+    with assms
+    have False by simp
+  } thus ?thesis by auto
+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
-
-lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
-  by (auto simp:s_waiting_def cs_waiting_def wq_def)
-
-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
+  using not_thread_holdents[OF assms]
+  by (simp add:cntCS_def)
+
+lemma cnp_cnv_eq:
+  assumes "th \<notin> threads s"
+  shows "cntP s th = cntV s th"
+  using assms cnp_cnv_cncs not_thread_cncs pvD_def
+  by (auto)
 
 lemma runing_unique:
   assumes runing_1: "th1 \<in> runing s"
@@ -2348,211 +3666,124 @@
   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)
+    unfolding runing_def by auto
+  from this[unfolded cp_alt_def]
+  have eq_max: 
+    "Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th1)}) =
+     Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th2)})" 
+        (is "Max ?L = Max ?R") .
+  have "Max ?L \<in> ?L"
+  proof(rule Max_in)
+    show "finite ?L" by (simp add: finite_subtree_threads)
+  next
+    show "?L \<noteq> {}" using subtree_def by fastforce 
+  qed
+  then obtain th1' where 
+    h_1: "Th th1' \<in> subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L"
+    by auto
+  have "Max ?R \<in> ?R"
+  proof(rule Max_in)
+    show "finite ?R" by (simp add: finite_subtree_threads)
+  next
+    show "?R \<noteq> {}" using subtree_def by fastforce 
   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
+  then obtain th2' where 
+    h_2: "Th th2' \<in> subtree (RAG s) (Th th2)" "the_preced s th2' = Max ?R"
+    by auto
+  have "th1' = th2'"
+  proof(rule preced_unique)
+    from h_1(1)
+    show "th1' \<in> threads s"
+    proof(cases rule:subtreeE)
+      case 1
+      hence "th1' = th1" by simp
+      with runing_1 show ?thesis by (auto simp:runing_def readys_def)
+    next
+      case 2
+      from this(2)
+      have "(Th th1', Th th1) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+      from tranclD[OF this]
+      have "(Th th1') \<in> Domain (RAG s)" by auto
+      from dm_RAG_threads[OF this] show ?thesis .
     qed
-    moreover have h2: "(?f ` ?B) \<noteq> {}"
-    proof -
-      have "?B \<noteq> {}" by simp
-      thus ?thesis by simp
+  next
+    from h_2(1)
+    show "th2' \<in> threads s"
+    proof(cases rule:subtreeE)
+      case 1
+      hence "th2' = th2" by simp
+      with runing_2 show ?thesis by (auto simp:runing_def readys_def)
+    next
+      case 2
+      from this(2)
+      have "(Th th2', Th th2) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+      from tranclD[OF this]
+      have "(Th th2') \<in> Domain (RAG s)" by auto
+      from dm_RAG_threads[OF this] show ?thesis .
     qed
-    from Max_in [OF h1 h2]
-    have "Max (?f ` ?B) \<in> (?f ` ?B)" .
-    thus ?thesis by (auto intro:that)
+  next
+    have "the_preced s th1' = the_preced s th2'" 
+     using eq_max h_1(2) h_2(2) by metis
+    thus "preced th1' s = preced th2' s" by (simp add:the_preced_def)
   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
+  from h_1(1)[unfolded this]
+  have star1: "(Th th2', Th th1) \<in> (RAG s)^*" by (auto simp:subtree_def)
+  from h_2(1)[unfolded this]
+  have star2: "(Th th2', Th th2) \<in> (RAG s)^*" by (auto simp:subtree_def)
+  from star_rpath[OF star1] obtain xs1 
+    where rp1: "rpath (RAG s) (Th th2') xs1 (Th th1)"
+    by auto
+  from star_rpath[OF star2] obtain xs2 
+    where rp2: "rpath (RAG s) (Th th2') xs2 (Th th2)"
+    by auto
+  from rp1 rp2
+  show ?thesis
+  proof(cases)
+    case (less_1 xs')
+    moreover have "xs' = []"
+    proof(rule ccontr)
+      assume otherwise: "xs' \<noteq> []"
+      from rpath_plus[OF less_1(3) this]
+      have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+" .
+      from tranclD[OF this]
+      obtain cs where "waiting s th1 cs"
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+      with runing_1 show False
         by (unfold runing_def readys_def, auto)
     qed
+    ultimately have "xs2 = xs1" by simp
+    from rpath_dest_eq[OF rp1 rp2[unfolded this]]
+    show ?thesis by simp
   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
+    case (less_2 xs')
+    moreover have "xs' = []"
+    proof(rule ccontr)
+      assume otherwise: "xs' \<noteq> []"
+      from rpath_plus[OF less_2(3) this]
+      have "(Th th2, Th th1) \<in> (RAG s)\<^sup>+" .
+      from tranclD[OF this]
+      obtain cs where "waiting s th2 cs"
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+      with runing_2 show False
         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:eq_waiting)
-      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:eq_waiting)
-      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
+    ultimately have "xs2 = xs1" by simp
+    from rpath_dest_eq[OF rp1 rp2[unfolded this]]
+    show ?thesis by simp
   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 card_runing: "card (runing s) \<le> 1"
+proof(cases "runing s = {}")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  then obtain th where [simp]: "th \<in> runing s" by auto
+  from runing_unique[OF this]
+  have "runing s = {th}" by auto
+  thus ?thesis by auto
+qed
 
 lemma create_pre:
   assumes stp: "step s e"
@@ -2581,649 +3812,34 @@
   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:
+lemma eq_pv_children:
   assumes eq_pv: "cntP s th = cntV s th"
-  shows "dependants (wq s) th = {}"
+  shows "children (RAG s) (Th 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
+    from cnp_cnv_cncs and eq_pv
+    have "cntCS s th = 0" 
+      by (auto split:if_splits)
+    from this[unfolded cntCS_def holdents_alt_def]
+    have card_0: "card (the_cs ` children (RAG s) (Th th)) = 0" .
+    have "finite (the_cs ` children (RAG s) (Th th))"
+      by (simp add: fsbtRAGs.finite_children)
+    from card_0[unfolded card_0_eq[OF this]]
+    show ?thesis by auto
 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)
+lemma eq_pv_holdents:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "holdents s th = {}"
+  by (unfold holdents_alt_def eq_pv_children[OF assms], simp)
+
+lemma eq_pv_subtree:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "subtree (RAG s) (Th th) = {Th th}"
+  using eq_pv_children[OF assms]
+    by (unfold subtree_children, simp)
 
 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:eq_waiting 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)
@@ -3272,7 +3888,7 @@
   { 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]
+    with tRAG_star_RAG
     have "(a, x) \<in> (RAG s)^*" by auto
     hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
   } thus ?thesis by auto
@@ -3288,7 +3904,7 @@
     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)
+    from tRAG_subtree_RAG 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 
@@ -3307,7 +3923,8 @@
         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)
+        hence "(Th th', x1) \<in> (RAG s)" 
+          by (cases, auto)
         then obtain cs where "x1 = Cs cs" 
               by (unfold s_RAG_def, auto)
         from rpath_nnl_lastE[OF rp[unfolded this]]
@@ -3359,19 +3976,46 @@
 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)
-  
+
+lemma dependants_alt_def1:
+  "dependants (s::state) th = {th'. (Th th', Th th) \<in> (RAG s)^+}"
+  using dependants_alt_def tRAG_trancl_eq by auto
+
+context valid_trace
+begin
+lemma count_eq_RAG_plus:
+  assumes "cntP s th = cntV s th"
+  shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+proof(rule ccontr)
+    assume otherwise: "{th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} \<noteq> {}"
+    then obtain th' where "(Th th', Th th) \<in> (RAG s)^+" by auto
+    from tranclD2[OF this]
+    obtain z where "z \<in> children (RAG s) (Th th)" 
+      by (auto simp:children_def)
+    with eq_pv_children[OF assms]
+    show False by simp
+qed
+
+lemma eq_pv_dependants:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "dependants s th = {}"
+proof -
+  from count_eq_RAG_plus[OF assms, folded dependants_alt_def1]
+  show ?thesis .
+qed
+
+end
+
+lemma eq_dependants: "dependants (wq s) = dependants s"
+  by (simp add: s_dependants_abv wq_def)
+
 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
+  using assms eq_pv_dependants dependants_alt_def eq_dependants by auto 
 
 lemma count_eq_RAG_plus_Th:
   assumes "cntP s th = cntV s th"
@@ -3382,6 +4026,113 @@
   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 inj_the_preced: 
+  "inj_on (the_preced s) (threads s)"
+  by (metis inj_onI preced_unique the_preced_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 holding_eq, unfolded cs_holding_def]
+  have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" 
+    by (unfold s_holding_def, fold wq_def, auto)
+  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)
+  { 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]
+        show ?thesis using vt_s'.unique_RAG by 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
 
@@ -3439,16 +4190,9 @@
     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")
@@ -3510,140 +4254,90 @@
   shows "(Th th) \<notin> Field (RAG s)"
 proof
   assume "(Th th) \<in> Field (RAG s)"
-  with dm_RAG_threads and range_in assms
+  with dm_RAG_threads and rg_RAG_threads 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
+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 eq_pv cnp_cnv_cncs
+  have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:pvD_def)
+  thus ?thesis
+  proof
+    assume "th \<notin> threads s"
+    with rg_RAG_threads 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 eq_pv_children[OF assms]
+      have "children (RAG s) (Th th) = {}" .
+      thus ?thesis
+      by (unfold children_def, auto)
+    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 -
+  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  show ?thesis using cnp_cnv_cncs by (simp add:pvD_def)
+  next
+    case False
+    with cncs_z and cnp_cnv_cncs show ?thesis by (simp add:pvD_def)
+  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
+
+context valid_trace
+begin
 (* ddd *)
 lemma cp_gen_rec:
   assumes "x = Th th"
@@ -3720,12 +4414,8 @@
   qed
 qed
 
-end
-
-(* keep *)
 lemma next_th_holding:
-  assumes vt: "vt s"
-  and nxt: "next_th s th cs th'"
+  assumes nxt: "next_th s th cs th'"
   shows "holding (wq s) th cs"
 proof -
   from nxt[unfolded next_th_def]
@@ -3736,9 +4426,6 @@
     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"
@@ -3771,17 +4458,91 @@
 
 end
 
--- {* A useless definition *}
-definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
-where "cps s = {(th, cp s th) | th . th \<in> threads s}"
-
-
-find_theorems "waiting" holding
+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)
+
 context valid_trace
 begin
 
-find_theorems "waiting" holding
+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")
+proof -
+  { fix th1
+    assume "th1 \<in> ?L"
+    from th_chain_to_ready[OF this]
+    have "th1 \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th th1, Th th') \<in> (RAG s)\<^sup>+)" .
+    hence "th1 \<in> ?R" by (auto simp:subtree_def)
+  } moreover 
+  { fix th'
+    assume "th' \<in> ?R"
+    then obtain th where h: "th \<in> readys s" " Th th' \<in> subtree (RAG s) (Th th)"
+      by auto
+    from this(2)
+    have "th' \<in> ?L" 
+    proof(cases rule:subtreeE)
+      case 1
+      with h(1) show ?thesis by (auto simp:readys_def)
+    next
+      case 2
+      from tranclD[OF this(2)[unfolded ancestors_def, simplified]]
+      have "Th th' \<in> Domain (RAG s)" by auto
+      from dm_RAG_threads[OF this]
+      show ?thesis .
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+lemma finite_readys [simp]: "finite (readys s)"
+  using finite_threads readys_threads rev_finite_subset by blast
+
+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)" (is "?L = ?R")
+proof(cases "readys s = {}")
+  case False
+  have "?R = Max (the_preced s ` threads s)" by (unfold max_cp_eq, simp)
+  also have "... = 
+    Max (the_preced s ` (\<Union>th\<in>readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)}))" 
+         by (unfold threads_alt_def, simp)
+  also have "... = 
+    Max ((\<Union>th\<in>readys s. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}))"
+          by (unfold image_UN, simp)
+  also have "... = 
+    Max (Max ` (\<lambda>th. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}) ` readys s)" 
+  proof(rule Max_UNION)
+    show "\<forall>M\<in>(\<lambda>x. the_preced s ` 
+                    {th'. Th th' \<in> subtree (RAG s) (Th x)}) ` readys s. finite M"
+                        using finite_subtree_threads by auto
+  qed (auto simp:False subtree_def)
+  also have "... =  
+    Max ((Max \<circ> (\<lambda>th. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})) ` readys s)" 
+      by (unfold image_comp, simp)
+  also have "... = ?L" (is "Max (?f ` ?A) = Max (?g ` ?A)")
+  proof -
+    have "(?f ` ?A) = (?g ` ?A)"
+    proof(rule f_image_eq)
+      fix th1 
+      assume "th1 \<in> ?A"
+      thus "?f th1 = ?g th1"
+        by (unfold cp_alt_def, simp)
+    qed
+    thus ?thesis by simp
+  qed
+  finally show ?thesis by simp
+qed (auto simp:threads_alt_def)
 
 end
 
 end
+